The neutron particle structure of the germanium isotopes

I I.E.I: 2.G

[

Nuclear Physics A202 (1973) 1--29; ~ ) North-Holland Publishing Co., Amsterdam

I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE NEUTRON PARTICLE STRUCTURE OF THE GERMANIUM ISOTOPES R. FOURNIER, J. KROON, T. H. HSU and B. H1RD

University of Ottawa, Ottawa, Ontario, Canada and G. C. BALL

Chalk River Nuclear Laboratories, Atomic Energy of Canada Ltd., Chalk River, Ontario, Canada, KOJ-1JO Received 2 October 1972 Abstract: The levels of 71'72'73"75Ge have been investigated with the (p, d) reaction at 20 MeV. DWBA predictions were used to obtain In transfers and absolute spectroscopic factors. In the reactions with the even-A targets only one In = 4 transfer was observed and a single In = 1 transfer to a low-lying level carried most of the available 2p½ spectroscopic strength. In contrast approximately 50~ of 2p~_ and I f~ spectroscopic strengths were found to be spread over many levels at higher excitation. The center-of-gravity energies and filling coetlicients of the 2pt, l f t , 2P~r and l g~ neutron subshells calculated from these measurements are in generally good agreement with the existing information on Ni, Zn and Se. The 2p~ and lf~ shells are almost full for N > 38 and additional neutrons preferentially fill the lg~ shell. E [

I

N U C L E A R REACTIONS :2.:a.7,,.76Ge(p ' d), Ep = 20.0 MeV; measured (7(0). 71.12.73.75Ge deduced levels, spectroscopic factors. Enriched targets.

1. Introduction

The experimental data available from the systematic studies of single-neutron transter reactions on the isotopes of Ni [ref. 1)], Zn [ref. 2)] and Se [ref. 3)], provide information on the behaviour of the centre-of-gravity energies and occupation probabilities of the active neutron shells outside the lf~ core as a function of the neutron number. No such systematic study has been reported for the germanium isotopes; the following analysis of the (p, d) reaction on 72, 73, :4, 76Ge ' together with our previous study of the 7°Ge(p, d)69Ge reaction 4) bridges the gap between Zn and Se. These measurements involve identifying the lo transfers to many levels in the final nucleus, so that detailed information on the excited states of the residual nucleus is obtained. Nuclei in the mass region of the germanium isotopes have so far eluded any reasonable model description. Full shell-model calculations have not been reported because of the large number of particles (or holes) outside the nearest closed core and the large number of available states. The even isotopes of germanium are poorly described by the vibrational model, and the calculations of Kisslinger and Sorensen 5), using the pairing Hamiltonian with a long-range quadrupole-quadrupole residual interac1 February 1973

2

R. FOURNIER et al.

tion, fail to reproduce the systematics of the low-lying levels of the odd-mass germanium isotopes. On the other hand, simple pairing theory is usually successful at predicting the occupation number and the centre-of-gravity energy (quasiparticle energy) of a particular shell;these depend more on the "bulk properties" of the nuclei, in contrast to the level energies which are sensitive to fine details in the nuclear wave function. Early investigations of 71Ge were carried out by studying the/?-decay of 7tAs and the 7°Ge(d, p)71Ge reaction 6). The high-resolution study of the 7°Ge(d, p)71Ge reaction by Goldman 7) using a multigap spectrograph showed a high level density at low excitations; spins and parities were assigned to the strongly populated levels. Measurements with the 7aGa(p, n) and the 71Ga(p, nT) reactions by Malan et al. 8) at energies between 1.5 and 3.0 MeV, using high-resolution neutron and 7-ray spectroscopy confirmed most of Goldman's levels and established several new ones below 2 MeV excitation with an accuracy of 0.5 to 1.0 keV. Recent measurements by Murray et al. [ref. 9)] of the 7-rays following the 7°Ge(d, p)71Ge reaction and the fl-decay of 71As confirmed the results of Malan et al. a) except that they report fewer energy levels. Recent high-resolution studies of the 7-rays following the fl-decay of 72As and 72Ga [refs. 10.11,12)] as well as proton 13) and deuteron 14) inelastic scattering experiments on 72Ge have extended the level scheme of 72Ge to approximately 4 MeV excitation. There is considerable information on the spins and parities of most of these levels. The main source of information on the energy levels of 73Ge comes from a highresolution magnetic spectrograph study of the 72Ge(d, p)V3Ge and the raGe(p, p') 73Ge reactions by Heymann et al. 15) who assigned spins and parities to the strongly populated levels. The only available information on the 75Ge level scheme is from the 74Ge(d, p)75Ge reaction and from the fl-decay of 75Ga [ref. 6)]. No spins and parities are assigned to these levels except for the recently measured 16) ½- ground state and the possibly -}+, 48 sec isomeric state at 139 keV [ref. 6)]. 2. Experimental method Isotopically enriched targets were bombarded with 20 MeV protons from the Chalk River MP tandem accelerator. A counter telescope consisting of three totally depleted silicon counters of thicknesses 50/~m ( A E ) , 700/~m(E), and 200 pm (E)was used to measure the energy, identify the type of particle and reject the numerous elastic protons ~7). Deuteron, triton and e-spectra were obtained simultaneously and stored in different regions of the CRNL on-line PDP1 computer memory. Spectra were accumulated at several angles between 15° and 65 ° with typically 30 to 40 keV resolution. An accurate energy scale for the deuteron spectra was established by making separate measurements of the 27Al(p, d)26Al reaction at regular intervals.

Ge ISOTOPES

3

The targets were made by evaporating the metal or the oxide, onto approximately 15 /.tg/cm 2 thick carbon foils, and their thicknesses which ranged from 20 to 85 //g/cm 2 were measured in a separate experiment by the Rutherford scattering of 6 MeV 0t-particles at 40 ° and 60 °. At these angles the elastic scattering peak from germanium is well resolved from that of the carbon backing and other light impurities. A single counter was used with the same geometry for the beam, the target and the counter slits as in the reaction measurements; this reduced the systematic errors in the calculation of the absolute cross sections. The target thicknesses determined from these measurements are listed in table 1 together with the isotopic enrichments of the material used in the evaporations. TABLE 1 Isotopic content and thickness of the germanium targets Target

72Ge age 7*Ge 76Ge

Isotopic content (~o) 73Ge 7*Ge

70Ge

72Ge

2.7 2.2 1.71 7.69

89.2. 7.6 2.2 6.65

2.2 78.0 0.91 1.69

4.3 11.1 94.5 10.1

76Ge 1.6 0.9 0.7 73.9

Target thickness Q~g/cmz) 36.8 84.4 19.4 19.5

A 3 mm silicon counter set at a constant angle of 20 ° to the beam direction monitored the elastically scattered protons from the target. A check at regular intervals of the ratio of the monitor counts to the beam charge established that no significant change occurred in the thickness of any target during the measurements.

3. Data analysis The positions of the observed peaks from the numerous 27Al(p, d)26A1 calibration runs were accurately determined with a peak-fitting program t s) and fitted to the wellknown 26A1 level scheme to yield a linear calibration. The slope of the calibration line was found to be almost constant and an average was taken. However, occasional discontinuous changes occurred in the intercept and these were corrected, for each spectrum, by requiring that the oxygen peak seen in all the spectra had the correct energy as determined from reaction kinematics. All the energy spectra of a particular reaction were transformed into Q-value spectra using the calibration and the kinematics of the reaction. The spectra were then summed and the peaks fitted with a peak-fitting program ~s) to determine their energies. After such a summation many weak peaks grew to a statistically significant size and a more accurate energy could then be assigned to weakly populated levels. Figs. 1-4 show the summed Q-value spectra for all reactions studied. The impurity peaks • Obtained from the Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA.

4

R. F O U R N I E R I

r

i

t--I

I

I

t

et al.

I

t

t

V 5

I

I

I

2000

72Ge ( p, d )

71Ge

(/)

x4

I-Z

0 u

8

IO00

, ! t5 [7 !

19

9~ I0 I

18 ,1~6 ~ J14

"

rIIL7

b, R vt[ il i,31 I£¢]_jH -12,0

-I 1.0

'; t! '

-I0,0 Q VALUE

~

~

-9.0

-8.0

( MeV )

Fig. 1. The summed Q-value spectrum of deuterons from the reaction 72Ge(p, d)71Ge. Impurity peaks are labelled by the residual nucleus and the peak number which appears in the corresponding Q-value spectrum of the (p, d) reaction on that impurity. i

[

[

l

~

~

i

I

T 1

il

I

I

I

I---T-

[

I

I

I

I

I

L

I

17

I

18

'

200C

16

!

14

•

I

73Ge ( p, d ) r2Ge

24

:

IOOC

i

! 2

l__1 - I0.0

I

J -9.0

I - &O

-7,0

- 6,0

I

I - 5,0

•

~ -

4.0

Q VALUE (MeV)

Fig. 2. The summed Q-value spectrum of the deuterons from the reaction 7aGe(p, d)72Ge. See caption for fig. 1.

Ge ISOTOPES L

i

[

3000

I

I

I

I

I

I

I

[

5 i

i

i

I

I

I

3

I

I

I

?4Ge ( p, d ) r3Ge x5

(/3 I-Z 2 0 0 0 0,,.)

'

~-

20 18

15 I ~

8

i I

I000--

9

3 4

II

l -~2.0

-~ LO

-~0.0 Q VALUE (MeV)

-9.0

-8.0

Fig. 3. The summed Q-value spectrum of deuterons from the reaction 74Ge(d, p)TaGe, see caption for fig. 1.

i

I

J

3000

I

J

I

[

l

I

I

I

T - ~ T - ~ ~ ~ - ~ - - ~

76Ge ( p' d ) "tSGe

I

t-2ooc

12

~

8

~"

~ i

~1~ iilili~ 17 1li151 13t +

- I LO

- I0.0

~

'

L

- 9,0

- 8.,

-7.0

- 6.0

Q VALUE (MeV) Fig. 4. The summed Q-value spectrum of deuterons from reaction 76Ge(p, d)?SOe. See caption for fig. 1.

6

R. F O U R N I E R et al.

are labelled by the residual nucleus and the peak number which appears in the corresponding Q-value spectrum of that impurity. The peak-fitting program used to analyse the deuteron spectra assumed a Gaussian line shape and allowed for the separation of many partially resolved doublets. In cases where this separation seemed doubtful because of poor statistics or too closely spaced levels, the energy of the individual peaks was retained but their angular distributions were summed. This situation occurred most often when the unresolved peaks were populatecl by different In transfers, e.g. In = 1 and 3, for which one angular distribution peaks where the other dips, and v i c e versa. By treating these doubtful cases in this manner, no spectroscopic strength was lost from an incomplete separation. TABLE 2 Optical-model potentials used in the DWBA analysis of the (p, d) reaction on the germanium isotopes

protons ") deuterons b) neutron

Vs (MeV)

ros (fm)

as (fm)

WD (MeV)

rol (fin)

al (fin)

46.1 105.1

1.25 1.07 1.25

0.65 0.962 0.55

13.7 20.25

1.25 1.366

0.47 0.668

The notation is that of Percy 2% a) Ref. 20). b) Ref. 21).

Since the targets were not isotopically pure and contained appreciable amounts o f the other germanium isotopes, it was necessary to make a correction to the peak area when a level from another isotope was known to overlap in Q-value with that peak. When 80 ~o or more of the observed intensity of a peak was accounted for by impurities it was assumed to be due only to these impurities. The DWBA analysis of the (p, d) reactions was made using the code D W U C K 19) which incorporates finite-range and non-local corrections into the conventional zerorange DWBA with local potentials. Figs. 5-8 show the experimental angular distributions for the (p, d) reactions compared with the theoretical curves calculated for the l, transfers that fit the data best. All the DWBA calculations used a finite-range parameter of 0.65 fm and assumed all the potentials to be local with no spin-orbit interaction and no lower cut-off. Different optical parameters taken from the literature were tried for both the entrance and exit channels and the combination that gave the best overall fit to our data is listed in table 2. The proton optical potential was taken from the work of Perey 20) (for 64Zn(p, p) at 22 MeV) and the deuteron potential was taken from the deuteron elastic scattering analysis of Percy and Percy 21) (for Cu(d, d) at 15 MeV). These parameters are the same as those used in our previous analysis of the vOGe(p' d)6 9 G e reaction 4).

Ge I S O T O P E S

7

The transferred neutron was assumed to be bound in a Saxon-Woods potential and the customary separation energy prescription was used. The radius and diffuseness of the Saxon-Woods potential were adjusted so as to have the best simultaneous agreement between the total extracted spectroscopic strengths for each reaction and the expected one based on sum rules while requiring, in addition, that the predicted shape of the angular distributions fit the experimental ones. A radius and diffuseness of 1.25 and 0.55 fm seemed to be the most appropriate values, and it was found that for a 10 ~ variation of the diffuseness, the cross sections varied by approximately 10 ~o, while a 5 ~ variation of the radius changed the cross sections by approximately 25 ~. For these small changes in the bound-state l~arameters, the shape of the predicted curves and the relative cross sections remained reasonably constant. This sensitivity to the bound-state parameters is reflected in the accuracy of the extracted absolute spectroscopic strengths. TABLE 3 S u m m e d spectroscopic strengths for la = 1, 2, 3 a n d 4 transfers Target

In = 1

In = 2

1, = 3

1. = 4

total

Expected total

72Ge

4.67

0.09

5.56

1.93

12.3

11.6

7aGe 74Ge

3.40 4.11

0.06 0.45

4.0-7.19 4.62

2.17-5.87 3.35

12.8-13.3 12;5

12.6 13.6

76Ge

3.75

0.47

5.36

4.2

13.8

15.7

Spectroscopic factors were extracted in the usual manner by fitting the theoretical angular distributions to the experiment, using the relation --~-/

= 2.29 ~

a~w(O),

where SP is the spectroscopic factor, a~n Dw(0) the predicted angular distribution and l. the transferred angular momentum. For unresolved doublets, the predicted cross sections for the two relevant in values were summed according to: aDw(0) = A C w ( 0 ) + (1 -A)a~?W(o), where A was varied between 0 and 1, and the resultant aDw(0) was fitted to the data in order to determine the best value of A and sometimes of It and/2. The total extracted spectroscopic strengths for the I. = 1, 2, 3 and 4 transfers seen in each reaction are shown in table 3 along with the total strength predicted by sum rules, which will be discussed in the next section. The deviations for 72Ge, 73Ge and 74Ge are all less than 8 ~. The larger (14 ~ ) discrepancy for 76Ge is attributable in part to the fact that the target used in the reaction measurements was destroyed and a different target was used in the thickness measurements with a subsequent renor-

8

R. FOURNIER et al.

malization, determined in a separate measurement of the (p, d) reaction at one angle, with an accuracy of approximately 10 ~. The addition of energy dependent terms in the real deuteron potential, as well as non-local corrections, was found to have little effect on the calculated absolute cross sections and they were therefore not included in the DWBA analysis since they would not significantly change the results of table 3.

4. Structure analysis From shell-model considerations, the (p, d) reaction in the germanium region is expected to pick up neutrons from the partially filled 2p~, lf~, 2p, and lg~_ subshells. In neutron pick-up from spin-zero targets, the spin and parity of the observed levels populated by an I, = 3 or 4 transfer can safely be assigned ~- and -~+ respectively; however, for In --- 1 transfers, the final spin can be either ½- or ~-. Different methods were attempted to distinguish between 2p½ and 2p~ neutron pick-up: (i) When (d, p) results are available, the ambiguity can sometimes be resolved by comparing the (p, d) and (d, p) reaction cross sections to a level populated by an In = 1 transfer. In this mass region where N > 38 the 2p~ neutron shell is almost full, while the 2p~ shell is just beginning to fill, so that the ratios of the (p, d) to the (d, p) reaction cross sections should fall into two different categories: those that are large, corresponding to stripping to, or pick-up from the 2p~ shell and those that are small, corresponding to a neutron transfer to or from the 2p~ shell. (ii) In each (p, d) reaction on the even germanium targets, one strong l, = 1 transfer was observed with a spectroscopic factor ~> 2; hence this transition must be a 2p~ neutron pick-up. (iii) From shell-model considerations, most of the 2P½ strength is expected to be at low excitations, while the 2p,} strength can appear at higher energies. In the case of the odd-mass 73Ge target whose ground state spin is 9 + , pick-up of an ( l , j ) neutron will lead to final states of parity (-)~ and spin J which are allowed by Clebsch-Gordan coefficient rules for the coupling of { a n d j to obtain J (except for lg~ pick-up, where only even-spin states can be populated). Further, usually more than one (/, j ) transfer can populate a level. The (p, d) reaction can populate both T< and T> states in the final nucleus. Using the Q-values of the (p, n) reactions and a Coulomb energy displacement of 10 MeV [ref. 22)], the calculated energies of the first T> levels in the germanium isotopes are approximately 9, 13, 10 and 12 MeV in 71, 72.73, 75Ge ' respectively, well above the highest excitation energy seen in the (p, d) reaction on these isotopes; therefore, only 7"< states were populated in these measurements. The spectroscopic strength will be divided between the 7"< and T> states and, according to a sum rule of Macfarlane and French 23), the total strength to the T< levels is ESj(T<) = ni-pJ/(2T+

1),

where nj and pj are the number of neutrons and protons in thej-subshell of the target nucleus and T is the isotopic spin of the target nucleus. Since the proton distribution

Ge ISOTOPES

9

in the germanium isotopes (Z = 32) is not known, the four additional protons outside the Z = 28 closed shell were assumed to be in the 2p~_ shell as predicted by the shell model. This assumption does not alter the total spectroscopic strength predicted for the T< levels. In addition, since T is large, it has only a small effect ( < l0 ~ ) on the spectroscopic strength predicted for the individual subshells. As a result, since p~ # 0 only for the 2p~ shell, the maximum possible spectroscopic strengths to the lf~, 2p~ and lg~ subshells are 6, 2 and l0 respectively; for the 2p~ shell, the expected total strength is: 3.56 for 72Ge (T = 4), 3.60 for 73Ge(T = 4.5), 3.64 for 74Ge (T = 5) and 3.70 for 76Ge(T = 6). The spectroscopic information extracted from this analysis is summarized in tables 4 to 7 which list the levels observed in these measurements, the transferred angular momenta, spins (even targets only), absolute cross sections and spectroscopic factors. 5. Comparison of the (p, d) data with other measurements 5.1. THE 72Ge(p,d)71Ge REACTION The experimental angular distributions obtained from the 72Ge(p, d)71Ge reaction are compared with DWBA predictions in fig. 5. The fits are satisfactory and allow unambiguous In assignments to be made for all the transitions. The energy levels of 71Ge populated in the present work are compared in table 4 to those seen in the 71Ga(p, n) and 71Ga(p, n~) reactions of Malan et al. s) and in the 7°Ge(d, p)71Ge reaction of Goldman 7). The level energies deduced from these different reactions agree within the experimental accuracy of each measurement. Peaks that were too weak to assign energies were seen between 502 and 706 keV, 807 and 1025 keV, 1595 and 1742 keV, 1966 and 2354 keV and above the 2354 keV peak; in addition, there was no evidence for the 60 keV level reported by Goldman. The spectroscopic information deduced from the present measurements is compared in table 4 with the results of the 7°Ge(d, p)71Ge reaction 7). Transitions to levels that have the same energy differ frequently in In assignment, probably due to the poor stripping patterns observed in the (d, p) work. However, the 1966 keV level must be a closely spaced doublet since a definite In = 0 transfer was observed in the (d, p) experiment and an unambiguous I, = 1 transfer is found in our data. Also shown in table 4 are the (p, d)/(d, p) ratios of the cross sections for the In = 1 transitions. Since the ground state spin of 71Ge is known to be 5-, a maximum ratio of 3.2 for transitions leading to ½- states was used to make the following spin and parity assignments: ~- to the 502 and 1288 keV levels, ~-, (5-) to the partially unresolved 1166+ 1210 keV doublet and to the 1595 keV level and (5-, ~-) to the 706 and 1096 keV levels. In addition the 1742 and 1966 keV levels were assigned g2-, (5-) since they are not seen in the (d, p) reaction and a spin of ~-, (5-) was assigned to the 2354 keV level since it is not likely that the 2p~ strength would spread to such excitations with a strength of 0.18. The Hauser-Feshbach calculations of Malan et al. a) for the 7XGa(p, n)71Ge reac-

2 I

1025 ~ 7 10964- 5

l +3

2

8074-10

1166±101 1210±10~

I 3

7064- 7 7574-10

9 l0

1

5024- 5

3

3+4

I

1864- 5

0

/n

7aGe(p, d)TIGe

energy (keY)

2

1

peak

~ - , (~t-) ~_

~+ 3 - c)

~+

~- c) 25-

.~•9 + ~-

~-

proposed j~r

0.15

0.05 0.72

0.05

0.45 0.055

5.0

1.20

3.05

(da/df2)m~x (mb/sr)

present work

0.05 0.40

0.03 0.37

0.03

0.17 0.31

3.64 1.93 2.32

1.04

S

1204.1 1212.4

1026.4 1095.8 I 139.4

808.1 831.2

708.1 747.1

174.8 197.8 499.7 524.3 588.7

0

890 950 970 1030 1090 1120 1160 1200

0 60 160 190 480 510 570 620 630 700 730 790 810

energy (keV) energy (keV) ( ~ --1 keV) ( ~ 4-10keV)

71Ga(p, n) a)

2? 27

1

(1)

(1) (2)

3 4 1 2

1

In

!

~j .~

7OGe(d ' p)71(

TABLE 4 The levels o f 7'Ge populated in the present work and comparison with other measurements

1410±10

15074- 5

15951 7

1742± 5 1786± 8

19664- 5

23544- 5

13

14

15

16 17

18

19

b) Re~ 7).

13544-10

12

~) Re~ a).

12884- 5

11

1

1

1 3

1

3

3

2

1

~) Re~. s.9).

~ - , (½-)

~-, (~-)

] - , (~-) ~-

~ - , (~-)

~-

~-

~+

~-

0.19

0.25

0.11 0.09

0.16

0.09

0.05

0.05

0.36

0.18

0.18

0.09 0.42

0.09

0.53

0.26

0.03

0.18

1965.4

1938.0

1743.5 1792.2

1.599.0 1629.4 1698.6

1506.5 1543.0 (1558.5) 1565.5

1414.5 1454.2 1476.5

1348.8 1378.9 (1406.5)

1288.3 1298.7

2350 2410 2480

1960 2040 2120 2170 2220 2270 2330

1780 1870 1940

1690

0 0

0 2

0 1 (3)

(1)

(3)

2?

1550 1590

(2)

(3)

0? (2)

1500

1410 1450 1470

1340 1380

1280

!

(~

(I

12

R. F O U R N I E R et aL

tion and the logft values deduced from the/~-decay of 7iAs by Murray et al. 9) yield a preferred spin of ~- for the levels at 502, 706 and 1096 keV; therefore the ambiguity in the spin assignment to the 706 and 1096 keV levels based on the (p, d)/(d, p) ratio

7ZGe(p,d ) 71Ge

g.s. o.o,/1

,.

~.

//~

t,'.

,~_

o.(

~}{ ~'

~_~~,..

1.742

o.7oo o., 1 =I

/

1.166

_

//~

0.807

[ I ] 0 20 40 60

/ ~ {

1o2 I

o.o, /

I

0

,.41=00.

I I I I 20 40 60 OC.M.

I

~14

0

t r I 20 40 60

I

r

Fig. 5. The angular distributions from the 7ZGe(p, d)7~Ge reaction compared with the predictions of the DWBA.

is resolved by these measurements. Furthermore, the ~-- spin assignment to the 502 keV level is in agreement with the systematics of the strong (S > 2 ) / , = 1 transfer discussed previously. Using the Lee-Schiffer effect 24-), Goldman 7) assigned a spin and parity of ½- to the 502, 706 and 1096 keV levels, in disagreement with the combined results of Malan et aL 8), Murray et al. 9) and the present analysis. All the/, = 1 transfers of Goldman have a dip at 140° except for those to the 1780, 2040 and 2730 keV levels; of these

Ge ISOTOPES

13

three, the 1780 and 2730 keV levels are weakly populated and their angular distributions have a p o o r 1, = 1 stripping pattern. In addition, the 2040 keV level is the third strongest 1, = 1 transfer observed by Goldman and has no dip at 140 °, implying a ~ - level which should also be strongly populated in the (p, d) reaction; however, it is

7aGe(p,d )72Ge -~T----r~~-r 0.1

3.754 5.055

:4

1.0

I+ 4 k ~ l : I +5 "~ ~.890

-

o.I

"~'~

0.850 .. ~~ .L: 2

\~1=

~ ~5.119

O.I

:1

0.1

o.o~ o]

1.455 " J.-- 2 oJ

0.,5

-

1.725 l =4 _

5.228 l= I 0.5

2.049

o.oI

--2

I~

0.1

2.505

0.1

0.1 -

1: I

O.OI

~ 2.956 \ ~ J_: 3

0 20 40 60

1.--3

5.965 1:1+3 047

,~ 1:1 ~

=1+4 _

V

.

1.6

~ /

+

~, 5.804

1=5

~5.598 _ ,~1=1+4 I~ :1+5

-~-" )" ~r% ,.554

~~1,.194 1=I

",~"~,

4.559

;

:1+5

I.O

659 "=1+40J : 1+5

0 20 40 60

"

4.458 1=1

0 20 40 60

8C.M. Fig. 6. The angular distributions from the ~aGe(p, d)~2Ge reaction compared with the predictions of the DWBA. too weakly populated in our work to assign an energy to it. Therefore, Goldman's data does not seem to display the Lee-Schiffer effect very well; better statistics and more experimental points in the region of the dip at 140 ° are probably required. 5.2. THE 7aGe(p, d)~2Ge REACTION The energy levels populated in the ~aGe(p, d)72Ge reaction are listed in table 5 and compared in fig. 9 to the level scheme of 72Ge based on the analysis of the ?-rays

14

R. FOURNIER

et aL

following the fl-decay of 72As and 72Ga by Camp x0) and Rester e t aL 11); there is overall good agreement for the excitation energies deduced from these different measurements. However, it is evident from fig. 9 that with the energy resolution of this work (30 to 40 keV) many of the levels populated in the 73Ge(p, d)TZGe reaction

7'4Ge(p. d ) 73Ge

2 . 4 "0

t.o

39

0.063

-

:

/~ \}y 1.139

_ T~r~ I.O

1.186

-- 1+3

3-

-

-

1.744 i

o.

~

-

0.494

,. o,

, °

0]"

.i

,.26l

-

3- = i

o.i

~ .557'

~X

t}{ 2.050 A

"

3.=1

/ ~ /~~'".122

_

}~

o.~

3-.,

J:2

-

~i

O,,I

0

o.I

~.312

0.897'

J. = I ,,

20 40 60

~

3_=I -

_

3.=I

0.1

]! }~ ;} o,t /

0

~2.27.0

1.610

)..: 3

20 40 60

2.329

0

20 40 60

~C.M. Fig. 7. The angular distributions from the 74Ge(p,d)TaGe reaction compared with the predictions of the DWBA. above 2 MeV excitation are most probably unresolved doublets. The position of the first 0 + excited state in 72Ge at 691 keV is indicated, in fig. 2, by the arrow labelled 0 +' and from the present measurements the transition to that level is estimated to be less than 0.5 ~ of the ground state transition strength. New levels were observed in 72Ge at 3119, 3228, 3398 and 3965 keV. Even though these levels overlap in Q-value with levels populated in the (p, d) reaction on the other germanium isotopes present

Ge ISOTOPES

15

in the target, these impurities account for only a fraction (< 40 7o) of the observed strength. In order to rule out the possibility that these new levels result from an incomplete impurity subtraction due to large errors in the manufacturer's quoted target composition, the 7aGe(p, t)71Ge spectra in which the ground state peaks from all the germanium isotopes are seen and well separated, along with the (p, t) reaction data "t6Ge( p, d ) "t 5Ge

o.tL ~ __ ~~712

"~'=I

1.0! f

'~- 0.19"7

o1~ . A ~

=l

O.iI

1.690

1 } ~ , ~'='1

~ L ,.o~- ..

0.574 , 3

0 20 40 60

I

Ilk

,.,o~

o.,~/~.

,=3

I

I

I

I

0 20 40 60

I

o,

~,

~.,o,

k f 3"=1+5

r/ ~,,'~, ~=,

I

0 20 40 60

I

I

Fig. 8. The angular distributions from the ~6Ge(p, d)TSGe reaction compared with the predictions

of the DWBA.

on the other germanium isotopes, were used to calculate the isotopic composition of the 73Ge target. The results are as follows: 2.9-3.6 ~ 7°Ge, 9.6-11.1% 72Ge, 73.477.7 % 73Ge ' 9.1-10.8 % 74Ge and 0.8-1 ~ 76Ge' in good agreement with those quoted by Oak Ridge (compare with table 1).

R. F O U R N I E R et al.

16

These new levels cannot be accounted for by the unassigned )'-rays observed in the fl-decay experiments of Camp ~o) and Rester et ak 11). However, it is possible that some of these levels are high-spin states which would be weakly populated in the fl-decay of 7 2 A s and 7 2 G a and also in the ),-decay from higher levels in 72Ge with low spins. TABLE 5 Summary o f the results o f the 7aGe(d, p)72Ge reaction 7aGe(p, d)72Ge Peak

1

excitation (keV)

2 3 4 5 6 7 8 9 10 11 12 13 14

0 690 830::[: 5 14554- 5 17234- 5 20494-10 25054- 5 2754 4-15 29364-10 3035 dz 15 3077=]=15 31194-15 3228 4-10 33164-10 33984- 5

15 16

3554±10 36594- 5

17 18 19 20 21 22 23

37544- 55] 38044- _, 38904-10 39654-10 4047-4-10 41944- 5 4339-4- 5

24

44584- 5

In

4 2 2 4 2 1 -}-4 1 3 3 1 1 1 1 +3 1+4 1 -}-3 1.}.3 1 -}-4 [1.}.3 !1+4 3 1 -}-3 1 -}-3 1 1.}.3 1 -}-4 1

(da/dD)ma~ (mb/sr)

S

0.34 < 0.002 0.21 0.07 0.09 0.03 0.84 0.21 0.04 0.06

0.52 < 0.003 0.04 0.01 0.23 0.01 0.19, 0.19 0.05 0.11 0.22

0.25 0.22 0.34 1.15 0.29 1.18

1.57 0.34 0.50 0.78 0.78 0.97 0.66

0.06 0.06 0.09 0.28, 0.28, 0.07, 0.32, 0.32, 10.38, [0.36, 1.23 0.15, 0.24, 0.27 0.37, 0.37, 0.27

0.50 0.60 0.18 1.28 1.41 0.90 1.14 0.50 0.53 0.51 0.55

Most of the DWBA fits to the experimental angular distributions shown in fig. 6 are found to be acceptable and allow many 1n transfers to be determined. However, it is not possible to distinguish unambiguously between In = 1 + 3 and In = 1 + 4 for the transitions to the 3398, 3659 and 4339 keV levels and to the 3754+3804 keV doublet. Since for these ambiguous cases the spectroscopic strengths of the In = 3 and 1n = 4 components do not differ significantly ( < 20 %), it is possible to estimate the total extracted spectroscopic strength by taking the average strength of S (In = 3) and S(ln = 4); this yields a total strength of ~ 10, which falls short of the expected

Ge ISOTOPES

17

raGe jw

~.~

E(keV)

( p,d )

(p,d)

I + (3,4)

4339

i

4194

I* 3 i*S i ~ (314) r1.

4047 3965 3690 3804 3764

1÷(3,4)

3659

ElkeVI ..IT

( I')-~ (r ................... (!*,3-) - /

(t,-b (e')'~

............... ................. ................

(1")

/4090 F6665 3685 ~"3664

i~

,l'J

,767 F3976 N67 3616 3560

(3~4") ~. L314.)

-3369

\'3318

2* S-

(i

~,,033"6

~'6341

(, ~ . ~ ,

E(keV)

AI,':3,o3 3766 3709 ~-3676 / ' 3 SS 6 35S0

(2,3,d~).% *

*

(f'.:'~'/

.............

6-J

I* I

3534

' ~ 6 4 19

I÷ {3,4)

~6~39 6s*l

,

33~6

I

3226

3094 30SS f1943 --2936

I 3

5116 3077 3035

3

2936

9734

I

9?54

3326

(1*,2*") E" 3~ ((-)~

6

3399

((1"} S') J ..............

3094 3036 ./'2930 ,%'~639 2943

..............

9764

(I,2,3)*

3" (4") (2")

ISiS 2444 240t

S" (3%4*) (~)

Z584 231 4 2463 2402

1+4

~506

3"

t064

3"

206 4

t

2046

4*

1728

4*

I?ti

4

1793

2*

1444

2"

1464

2

14 66

2*

834

2"

934

Z

830

O*

S91

O*

960

O*

O

O*

4

0

(I.2)

D.C.Camp

10)

.............

-

Resteret.

-

ol.

O Ill

Present

measurements

Fig. 9. The energy levels of ~2Ge obtained from the ~3Ge(p, d)~2Ge reaction compared with the levels deduced from the analysis of the y-rays following the fl-decay of V2As and V2Ge by Camp lo) and Rester e t al. 1]).

18

R. F O U R N I E R et al. "~

+

+

I

O0

~

~

"t-

=

o

E

m

~.~

¢=

~,1

+

~

,

¢,1

~I

I

÷,

C,I

¢'l

. .

t".l

~

.

GelSOTOPES

19

a

- - ~

~

~

~

20

R. FOURNIER et al.

value of 12.6. The non-negligible strength to the numerous excited states above 4.5 MeV excitation was taken into account by summing the area from 4.5 to 5.5 MeV excitation in each spectrum; the angular distribution obtained is shown in fig. 10. It is best fitted by a mixture o f / . = 1 + 3 + 4 with spectroscopic factors of 0.6, 1.2 and 1.2 respectively. The total spectroscopic strength then becomes 13 which is close to the sum-rule value.

I

3.o

73Ge(p,d)7~Ge

/ 2.0

t

.~

1.0 0.8

/

'

0.6

0.4 0.3

0

20

40

60

e c.ffl,

Fig. 10. The angular distribution of many unresolved levels between 4.5 MeV and 5.5 MeV excita tion compared with the DWBA assuming a mean excitation of 5 MeV and mixtures o f l . = 1, 3 and 4 transfers. 5.3. THE ~4Ge(p, d)TaGe REACTION The spectroscopic information extracted from the 74Ge(p, d)TaGe reaction is compared in table 6 with the results of Heyman etal. 15) obtained from the 72Ge(d, p)73Ge and the 73Ge(p, p')V3Ge reactions; the last two columns of the same table list the energy levels and some possible J~ assignments taken from ref. 6). The measured excitation energies of levels seen in the three reactions agree within the experimental accuracy of each measurement, but the assigned angular m o m e n t u m transfers often conflict with one another, possibly suggesting that some of these levels are unresolved doublets. The D W B A predictions fit most of the experimental angular distributions as can be seen in fig. 7. The first angular distribution corresponds to the unresolved ground state (7o + , 1. = 4) and 13.5 keV level (~+, 1. = 2), while the third one corresponds to the unresolved 368 and 396 keV doublet of Heyman et al. The angular distribution

Ge ISOTOPES

21

of the 494 keV level is best fitted with a mixture of I. = 1 + 2. Although the fit is still poor, it seems to imply the existence of an unresolved doublet, which is not in contradiction with the work of Heyman et al. since this level was populated by an In = 1 transfer in their (d, p) reaction and by a AL = 2 transfer in the (p, p') reaction. In addition, the (d, p) angular distribution to this level does not appear to be a pure In = 1 transfer. The In = 3 assignment to the 666 keV level seen in the (d, p) reaction seems questionable since this level, which should be more strongly populated in a (p, d) reaction, was not seen in our spectra. It is also in disagreement with the AL = 2 transfer to that level observed in the (p, p') reaction. Furthermore, the published (d, p) angular distribution for this transition is almost identical to that of the ground state, suggesting it could be an In = 4 transfer which would agree with the (p, p') data and our results; it should then be only weakly populated in our reaction since the lg~ shell is only partially filled and most of the In = 4 strength goes to the ground state. The 1261 keV level which has an unambiguous I, = 1 stripping pattern in the present work must be different from the 1274 keV level of Heymann et al. which is clearly observed as an In = 0 transfer in the (d, p) reaction. Similarly, the levels at 1610 keV (In = 3) and 1653 keV (In = 1) must be different from the 1617 keV (ln = 2) and 1646 keV (/n = 2) levels of Heymann et al. since they are populated so strongly in the (d, p) work. Several weak and partially unresolved levels were seen in the deuteron spectra between the 1744 and 2030 keV levels. There was, however, no evidence for the l n = 1 transitions to the 1936 and 1983 keV levels which were strongly populated in the (d, p) reaction and should be populated even more strongly in a (p, d) reaction if they are {- states. Table 6 also lists the (p, d)/(d, p) ratios for the l n = 1 transitions, which predict spin assignments of ½- for the 63 keV, 494 keV (for the In = 1 component) and 897 keV levels and a spin o f { - for the 370 and 1039 keV levels. The other levels populated by l n = 1 transfers were assigned ½-, (½-) since these levels are either not seen or very weakly populated in the (d, p) reaction. Even though the (p, d)/(d, p) ratio suggests a spin of ½- for the 2030 keV level, this level was assigned ½-, (½-) since a 2p½ pick-up strength of 0.18 is unlikely at this high excitation energy. 5.4. THE 76Ge(p, d)7SGe REACTION The energy levels of 75Ge populated in this work are compared in table 7 with those reported previously 6), mainly from a study of the 74Ge(d, p)75Ge reaction. For levels below 1 MeV, the measured energies agree reasonably well with one another but above 1 MeV there is little agreement between the two level schemes. This is probably because In = 0 and 2 transitions are favoured in the (d, p) reaction, while the (p, d) reaction preferentially picks up neutrons from the almost full 2p~ and lf~ shells. The angular distributions obtained from this reaction are shown in fig. 8, together with the DWBA predictions. Unambiguous I, assignments can be made to all the transitions with the exception of that to the 1690 keV level which is seen to be poorly

22

R. F O U R N I E R et aL

fitted by an In = 3 transfer. Although this level seemed to be a closely spaced doublet in the energy spectra, mixtures of different l, values did not improve the fit significantly. TABLE 7 Summary of the results of the 76Ge(p, d)VSGe reaction Peak

Excitation (keV)

1

0

In

0.61

0 62 139 180 215

½- b)

1.2

½% ~ ~-

0.65 0.55

0.16 1.38

1]-~-

0.12

0.32 1.59

5

457+ 5

6

574+ 5

3 _tl 3 2 1

12504-15

2.60

0.42 4.2

1 3

10

j~r

~-+ ~+

2504- 5 314! 5

9884-10

Excitation a) (keV)

+42 3 4

9

S

½-

197+ 5

671±15 8894-10

(dcr/df2)ma~ (mb/sr)

1

2

7 8

Proposed j~r

6.0 ~~+ ½% ~ -

0.15 0.48

261 326 360

593 1.59 0.05 0.14

685 885 940 1110 1320

I1

14034- 5

0.62 3

~--

0.47 1470

12 ¢) 13 14 c)

1501 4- 5 1593+10 1690± 5

1 3 3

~-, (½-) ~~-

1.3 0.18 0.12

0.48 0.64 0.51

15 c)

18034- 5

1

~-, (½-)

0.60

0.22

~-, (~-)

16 17

2043 ± 10 2105±10

1 + 3

18 19

2198±15 23164-10

1

~-, (½-)

0.45

0.18

20

26704-10

1

{-% (½-)

0.19

0.11

1570 1740 1870

0.06 0.16

~-

2080 0.45 2190 2230

a) Ref. 6). b) Ref. x6). 0 Broad peaks, possibly multiplets.

(2460) 2680 (2830) 3020

(~-+) ~)

Ge ISOTOPES

23

The strong ( S = 1.59) I. = 1 transition to the 574 keV doublet allows us to assign a ½- spin to one of the m e m b e r s o f this doublet. Probable spins o f ½-, (½-) were assigned to all the levels above 1 MeV populated by In = 1 transfers since the 2p~ strength is not expected to fractionate into levels very m u c h above that energy.

:n `i

o

"*~

I

~a,,.Z

"i i ,

.tn=4 (9/~l

1n=3(5,~)

~

o

z

o

i

~

o

i

i

Y3Ge O,

,I

o

,

il ,

o

o,. o

~

,,

I.i

i

l

. '

,.... z.

,,

~

z

II o

ifl ,

I

'

'

,

,

,!

o

,

~,

,I

o ~ EXCITATION ENERGY (MeV)

?f, o

o

,.

i

i

~,

.

Fig. 1l. The distribution of the spectroscopic strength to the levels populated by lo = l, 3 and 4 transfers in the 7°,72"74,76Ge(p, d)69.71"Ta'TsGe reactions. For the In = 1 transfers, the heavy lines represent ½- levels, the thin lines represent {-- levels and a question mark above a line represents an ambiguous ½-, {- assignment.

6. Discussion

The results of the (p, d) investigations on the even germanium isotopes are summarized in fig. 11 which shows the spectroscopic strength distribution in each residual nucleus. On the horizontal axis is shown the excitation energy in the residual nuclei, and on the vertical axis is plotted the spectroscopic factor for the three transferred 1n values of interest. The results f r o m our previous analysis of the 7°Ge(p, d)69Ge reaction 4) are included for completeness. It can be seen that the !~ = 1 transfer is characterized by two strong transitions at low excitations; one at ~< 100 keV excitation which carries most of the observed 2p~r strength and a much stronger transition at approximately 500 keV, corresponding to

24

R. F O U R N I E R et aL

50 % of the available 2p~ spectroscopic strength. The ln = 3 transfer is characterized by a single strong transition, with ~ 50 % of the observed I f l strength to a ~ state below 500 keV; the only exception is in 75Ge ' where the strength is divided among three ~- levels near 500 keV excitation. Only one In = 4 transition is observed to each residual nucleus and it carries most of the available lg~ spectroscopic strength. TABLE 8a Occupation ( Vfl) Nucleus 7°Ge 72Ge 74Ge 76Ge

2p~v

2p~

2p½ + 2p~

0.30-0.49 0.49 0.45 0.36-0.52

0.69-0.80 0.96 0.98 0.87-0.97

0.62 0.84 0.79 0.76

1f'l-

I g~_

0.86 0.87 0.84 1.0

0.10 0.18 0.37 0.48

TABLE 8b Center-of-gravity energy (MeV) Nucleus 69Ge 71Ge 7aGe 75Ge

2p~v 0.09-0.42 0.0 0.30 0.0 -0.18

2pk

1f~

I g-I-

0.57-0.62 0.84 0.81 1.05-1.11

0.55 0.60 0.80 0.93

0.40 0.19 0.0 0.20

The filling coefficients Vj for each subshell were obtained from the (p, d) data with the relation V 2 -- N Z S ; xp ' S~ "~ where S~.~x is the maximum possible strength to shell j, ~ S~~p is the observed total strength to that shell and N is the constant which normalizes the total extracted spectroscopic strength to the sum-rule value for each reaction. The center-of-gravity energies for each subshell were deduced from _ ZS~Ej '

where Sj is the spectroscopic strength to the level of energy E j . These two quantities are listed in table 8 and compared, in fig. 12, to other available data in this mass region; the Ni data was taken from the work of Turkiewicz et al. and Cosman et al. x), the Zn data from the work of Von Ehrenstein et al. 2) and the Se data from the work of Lin 3). The curves represent the results of pairing calculations to be discussed later. It is evident from these data that the 2p~ and If;, subshells

Ge ISOTOPES

25

are very closely spaced and fill simultaneously for 30 < N < 38. The 2p~ and lg~ subshells lie at a higher energy and also fill simultaneously for 38 < N < 50. It was hoped to detect the effect of the short-range p-n force by comparing the center-of-gravity energies of the 2p~ and lf~ shells for even-Z isotopes in this mass region. In particular, the addition of two protons (mainly 2p~ or lf~) in going from one isotone to another should cause the center-of-gravity energy to shift to higher , Ni (d,p) I ' '

L ~

3I~"

, I

,

o Z n ( d , p)

~

o Ge(p,d)

g"

" Se(d,p)

,

Ig, /.

:'

2p,,

0

,

2%/ 0.5

~ o

v

°

o

°

"

"

0.5

0

0

I o _1

0

i

0

I

40 NEUTRON NUMBER

50

50 NEUTRON NUMBER

Fig. 12. The single-particle energies and the occupation probabilities of the 2p~, lf{, 2p~. and lg{ shells for some nuclei in the region 28 < N < 48 compared with the results of the pairing theory (see

text).

excitation. However, from the present results it must be concluded that either this type of data is not accurate enough to detect the effect or that the short-range p-n force is weak in this mass region.

26

R. F O U R N I E R et aL

Also of interest are the observed I, = 2 transitions to levels in 71, '/2, 73, 75Ge with respective total spectroscopic strengths of 0.09, 0.06, 0.45 and 0.47. These two latter values may not be very reliable since ~ 50 ~ and 90 ~ of the I, = 2 strengths to 73Ge and 75Ge, respectively, come from l~ = 2 + 4 fits to transitions which are predominantly l~ = 4, whereas no pure l~ = 4 transfers were observed in this region of Q-value to compare with the DWBA calculations. In addition, ~ 25 ~o of the In = 2 strength to "/aGe c o m e s from a rather poor In = 1 + 2 fit to a level in 73Ge at 494 keV. Nevertheless, these results indicate that the 2d~ shell is starting to fill even though the major N = 50 shell is not closed. Previous results are in disagreement as to how much the 2d~ configuration mixes into the ground state of nuclei in this mass region. Lin 3) in the study of the 78, 80, 82Se(d ' t) reactions observed In = 2 transitions with total spectroscopic strengths of ~ 0.6 in each reaction, while Bercaw et aL 25) reported l~ = 2 transfers with a total strength of ~ 0.1 in the SSSr(d, t) reaction and Ball et aL 26) report an In = 2 spectroscopic strength of 0.05 in the 9°Zr(p, d) reaction. 6.1. PAIRING CALCULATIONS

Pairing calculations were made for the region 30 < N < 48 in order to see how successful simple pairing theory would be at predicting the behaviour of the occupation and the centre-of-gravity energy of each shell as a function of N. The gap equations 2'/) were solved using a pairing force strength parameter of G = 24/.4, the value recommended by Kisslinger and Sorensen s) for this mass region. Only the 2p~, lf~, 2p,r and lg~ shells were considered, and within each shell the single-particle energies were given a smooth A-dependence of the same form as that used by Kisslinger and Sorensen 5). The occupation V 2 and the single-quasiparticle energies Ej were calculated from the relations , / ( ~ j - ~ ) 2 + A2_I' Ei = x / ( e j - 2 ) 2 + A 2 - - c o n s t ,

(1)

where the ej are the single-particle energies used in the calculations, and 2 and A are determined by solving the gap equations. The constant in eq. (1) was set equal to - A, which corresponds to setting Ej = 0 for the single-particle state which is half-filled. The calculated V z and Ej were then compared with the experimentally measured values of the occupation and the centre-of-gravity energies, respectively. A first calculation using the recommended single-particle energies of Kisslinger and Sorensen 5) for 58Ni (i.e., ~0p~_ = /~0f.,~ = 0, e0p42. = 4.0 MeV and e°lsl = 3.0 MeV), overestimated the observed separation of the 2P½ and the lg~ shells. Using different o ~ ~ e°f~ ~ 0 and single-particle energies for SSNi, it was found that values of e2p e°p~ ~ e°g~ ~ 3.0 MeV yielded satisfactory fits to the centre-of-gravity energy and

Ge ISOTOPES

27

the occupation of the four subshells. In fig. 12, the curves correspond to the set o ½ = 2.8 MeV and ~ o = 3.0 MeV. e°p~ = 0, et°ft = 0.3 MeV, 82p The large discrepancies observed for the center-of-gravity energies of the 2p~ and lf~ shells in the selenium isotopes can be attributed to the fact that only one or two low-lying In = 1 and 3 transitions were observed in the (d, p) study of Lin 3) while the present (p, d) investigation shows that the 2p~ and lf~ strength spreads considerably to high excitation. It is evident that a detailed study of single-nucleon pick-up strengths from the almost full shells in the selenium isotopes is required. The gap energy A which is determined by solving the gap equations can also be evaluated from the even-odd mass difference, using the relation: A ~ ¼ [ I S E ( N ) - S E ( N - 1)l + I S E ( N ) - S E ( N + 1)l], where N is the number of neutrons in the nucleus for which A is calculated, and SE(N) is the separation energy of one neutron. Using the tabulated values of the mass excesses, A was calculated to be approximately 1.5 MeV in this mass region, and a value of approximately 1.3 MeV was obtained by solving the gap equations. Therefore, the strength of the pairing-force parameter is adequate for this mass region since the value of the gap energy A is sensitive to the chosen value of G. It can be concluded from these results that simple pairing theory reproduces very well the systematics of the single-particle energies and the filling coefficients of the shells in this mass region. 6.2. SHELL-MODEL CALCULATIONS Since it is found that the 2p~ and lf,2 shells are almost filled for all germanium isotopes with N > 38 (see table 8), it should be possible, to first order, to write the main components of the even target ground state wave function as follows: 7O+,Ge = Otn(lg?r)n+fln(lg~)n-2(2p~) 2. In addition, since the transitions to the lowest ½- and {+ levels in the odd isotopes carry most of the observed 2p½ and lg,, spectroscopic strength, respectively, the wave functions for these states are given, to a first approximation, by 7°+n-lGe~r_ = (lg~)n-2(2P½), 7°+n-aGe{+ = ot,_l(lg~)"-l+fl,_t(lg~)"-a(2P½) 2, 2

2

2 where ~, ~ or,_ 1, fin2 ~ fin-x and n is even. The measured spectroscopic factors to the ½- levels can be used to determine the amplitudes of the (I g~)" - 2(2p½)2 configuration in the even target isotopes; specifically:

f12 = ½S2p½ = 0.49,

fl] = 0.31,

f12 = 0.36.

If we make the simplifying assumption that ot2 = ~2 = ot2 = 0.6 and//22 = f12 = f162 = 0.4, then the spectroscopic factors for Ig~ neutron pick-up can be written as:

S(In

= 4) = [0.6~/n (g~l)g"~-~)___0.4~/n-2(g~-21}~-a)] 2,

28

R. F O U R N I E R et al.

where the symbols (I}) are the coefficients of fractional parentage, n is replaced by n - 1 for an odd target nucleus and the sign of the second term is determined by the signs of ~tn/flnand 0tn-1/fl~-x t. This equation predicts maximum values of 5.2, 3.1, 1.2 and 0.88 for the 1, = 4 transfers to the lowest ~+ levels in 75Ge, 7aGe and 71Ge and to the ground state of 72Ge respectively. These predictions are in good agreement with the observed (normalized) spectroscopic factors of 4.8, 3.7, 1.8 and 0.52, respectively. Furthermore, for even target nuclei, these predicted spectroscopic factors represent > 95 ~o of the total lg~ pick-up strength and are consistent with a (p, d) transition to only one ~+ level in the odd nuclei. If we assume that the anomalous first excited 0 + state in 72Ge has a wave function which is orthogonal to that of the ground state 2s), then the present model can also account for the very weak (p, d) transition to this level. In particular, the spectroscopic factor, which results from the cancellation of two nearly equal amplitudes, is predicted to be ~ 0.0027 compared with the experimental value of ~ 0.003. In the 7aGe(p, d)72Ge reaction, pick-up of algcr neutron can also populate states in 72Ge having configurations (lg~r)22 +, 4% 6 +, s + with predictable spectroscopic factors of 0.15, 0.27, 0.39 and 0.51, respectively. The 2 + and 4 ÷ states are expected to mix strongly with the low-lying collective levels in 72Ge while the 6 + and 8 + levels should remain relatively pure. The two known 2 + levels at 830 and 1455 keV in 7 2 G e are populated by almost pure 1, = 2 transfers (see fig. 6). The addition of In = 4 components to these two angular distributions does not improve the fits noticeably and the combined maximum spectroscopic strength of ~ 0.06 for the In = 4 components, which still give acceptable fits, is only ~ 40 ~ of the predicted value. Two In = 4 transfers were observed to the 4 ÷ levels at 1723 and 2463 keV with a combined spectroscopic strength of 0.42, which is to be compared to the predicted value of 0.27. Other possible In = 4 transfers were observed to levels between 3 and 5 MeV with strong spectroscopic factors consistent with those expected for 6 ÷ and 8 ÷ levels but the ambiguity in the In assignments to these levels prevented us from locating these high-spin states. t Because of the arbitrariness of the phases of the coefficients of fractional parentage, it is not possible to determine the signs of cx,,/fl,, and ~n-l/fl,,-1.

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Ge ISOTOPES 8) 9) 10) 11)

12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)

29

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