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The (p, t) reactions on nuclei in the rare earth region
I 2.B:2.G I I
THE
Nuclear Physics A213 (1973) 221 --266; (~) North-Holland Publishiny Co., Amsterdam N o t to be r e p r o d u c e d b y p h o t o p r i n t or microfilm without written permission f r o m the publi8her
(p, t) R E A C T I O N S
ON NUCLEI
IN THE
RARE
EARTH
REGION
MICHAEL A. O O T H O U D T t and N O R T O N M. H I N T Z J. H. Williams Laboratory, School of Physics, University of Minnesota, Minneapolis, Minnesota 55455tt Received 12 March 1973 (Revised 14 June 1973) Abstract: The (p, t) reactions on isotopic targets of 17s. 1s OHf and all the stable isotopes of Yb and on natural targets of Gd, Dy, Er, HI', Ta, W, Os and Au were studied at a beam energy of 19 MeV with an average resolution of 12 keV. A split-pole magnetic spectrometer was used to measure (13,t) Q-values and absolute differential cross sections. On the basis of angular distribution shapes definite 0 + and tentative 2 + assignments were made. Rotational bands were identified assuming an I ( I + 1 ) spacing. The (p, t) reaction populates excited 0 + states strongly in 174yb, 1 7 6 H f , 166yb and several Gd, Dy and Er isotopes. The 174yb and 176Hf 0 + states are discussed in terms of the pairing phase transition and in terms of Nilsson orbitals with unequal (p, t) reaction amplitudes. Members of gamma and octupole vibrational bands were observed in the even-N nuclei. The lowest L = 0 transfers to states in 169,171yb were found to have less than 55% of the strength to ground states in adjacent even-N nuclei. A strong L = 0 transfer to a state at 1513 keV in 171yb indicates the presence of a possible K = 0 core vibration coupled to the unpaired ~}[512] neutron. The natural targets have furnished information on trends in cross sections for members of ground bands, gamma bandheads, 3octupole states, and strongly excited 0 + states. N U C L E A R REACTIONS 168. 170,171,172. 173. 174-, x76yb ' 178, lSOHf' Gd, Dy, Er, HI', To, W, Os, Au(p,t), 176yb(p,p'), E = 19 MeV; measured ¢r(Et, 0), Q. E
1 6 6 ' 1 6 8 ' 1 6 9 " 1 7 ° ' 1 7 1 ' 1 7 2 " 1 7 4 y b , 1 7 6 " 1 7 8 H f d e d u c e d levels, J , 917. I 1 2 , 1 1 3 , 1 1 4 , 1 1 6 C d , 154, 155, 156, 157, 1 5 8 , 1 6 0 G d ' 160, 1 6 1 , 1 6 2 , 1 6 3 , 1 6 4 D y ' 1 6 4 , 1 6 6 , 1 6 7 , 1 6 8 , 1 7 0 E r ' 1 7 6 , 1 7 7 , 1 7 9 H f ~
lSlTa ' 182.183,184, 186W, 188,1s9,190,1920s ' (p, t), E = 19 MeV; measured Q. Enriched and natural targets.
1. Introduction T h e (p, t) r e a c t i o n o n d e f o r m e d a n d t r a n s i t i o n a l nuclei in t h e r a r e e a r t h r e g i o n has b e e n t h e s u b j e c t o f several r e c e n t e x p e r i m e n t a l i - 8 ) a n d t h e o r e t i c a l 9 - 1 3 ) studies. T h e m a j o r p o r t i o n o f t h e (p, t) r e a c t i o n s t r e n g t h f o r d o u b l y e v e n nuclei t y p i c a l l y g o e s to m e m b e r s o f t h e g r o u n d state r o t a t i o n a l b a n d w i t h o t h e r states b e i n g p o p u l a t e d w e a k l y . I n t h e S m nuclei, h o w e v e r , e x c i t e d 0 + states a s s o c i a t e d w i t h t h e c h a n g e in e q u i l i b r i u m d e f o r m a t i o n n e a r N ( n e u t r o n n u m b e r ) -- 88 h a v e b e e n o b s e r v e d w i t h o v e r 57 % o f t h e g r o u n d state s t r e n g t h 1). I n t h e G d , D y a n d E r nuclei 0 + states w i t h a p p r o x i m a t e l y 15 % o f t h e g r o u n d s t a t e s t r e n g t h h a v e b e e n r e p o r t e d 4 - 6 ). S u c h states m i g h t be a s s o c i a t e d w i t h g a p s in t h e single-particle s p e c t r u m 2) o r w i t h t Supported in part by the National Aeronautics and Space Administration and the National Science Foundation. Present address: Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540. t* Supported in part by the US Atomic Energy Commission. This is report no. COO-1265-131. 221 October 1 9 7 3
222
M . A . OOTItOUDT AND N. M. HINTZ
a nonuniform distribution of Nilsson orbitals with oblate and prolate shapes near the Fermi surface as in the actinide region 14). The present work was undertaken to investigate the (p, t) reaction on Yb and H f isotopes. These nuclei have large (fl ~ 0.3) stable deformations 15) and have an approximately uniform distribution of oblate and prolate orbitals near the Fermi surface. As we have previously reported 2), strong transitions to excited 0 + states have been found which may possibly be attributed to a local gap in the Nilsson TABLE 1 Composition of enriched Yb and H f targets a)
Target
i76yb i74yb IVayb i72yb iVlyb iT°yb 168yb
Isotopic abundance (%) 176yb
t74yb
i7ayb
i72yb
17iyb
170yb
i68yb
96.43 0.57 0.52 0.28 0.20 0.73 6.18
2.12 95.8 10.2 2.02 0.91 3.1 20.22
0.56 2.20 85.1 3.46 0.63 2.3 12.30
0.52 0.99 3.40 91.5 2.03 4.8 19.85
0.32 0.41 0.66 2.57 95.96 7.8 17.27
0.05 0.05 0.16 0.15 0.27 81.4 5.89
< < < < < <
Target
180Hf 178ttf
0.01 0.01 0.01 0.01 0.02 0.07 18.25
Isotopic abundance (%) lsoHf
i79Hf
178Hf
i77Hf
i76H' f
i74Hf
98.21 3.07
0.80 2.90
0.65 89.14
0.27 4.36
0.07 0.52
< 0.03 < 0.05
") As measured by the supplier, Isotopes Division of Oak Ridge National Laboratory.
energy level spectrum. In this work we discuss in more detail the L = 0 transitions in the H f and Yb isotopes and report results for population of 7-vibrations, octupole vibrations and other states in both e v e n - a n d o d d - N nuclei. In sect. 2 we present the experimental procedure. In sect. 3 proton scattering data for 176yb and the results of an optical-model analysis are given. Data for the (p, t) reaction on isotopic targets of 178Hf' ~80Hf and all the stable Yb isotopes are given in sect. 4. [Tabulated cross section data for the (p, p) scattering and (p, t) on isotopic targets may be found in ref. 8) or obtained from the first author (M.A.O.). ] We also present the results of a survey done with the (p, t) reaction on natural Gd, Dy, Er, Hf, W, Ta, Os and Au targets. In sect. 5 we present the results of a DWBA analysis of the t 76yb(p ' t) data and compare to recent theoretical analyses 9=13).
2. Experimentalprocedure A 19 MeV proton beam was obtained from the Williams Laboratory Tandem Van de Graaff Accelerator. Defining slits at the image of the analyzing magnet were set to give a 5 keV spread in beam energy. Data were taken using an Enge split-pole
223
(p, t) REACTIONS
TABLE2 The L = 0 cross sections and Q-values for (p, t) on natural targets Other experiments
This experiment Target
a(27½ °) (/zb/sr) ")
Qg.s. (MeV) b)
S2n c) (MeV)
Szn (MeV) ~)
Ref.
1 1 1 3
--7.8914--7.4564--7.1064--6.3634-
5 5 5 5
16.373 15.938 15.588 14.845
16.375415.938___ 15.586414.837±
3 3 3 3
71) 71) 71) 71)
154Gd 155Gd 156Gd 157Gd 158Gd 16°Gd
~ 4064-41 <~ 2784- 16 ~ 7134- 18 <~ 4554- 16 7144- 16 6164- 16
--6.6604--6.8564--6.4904--6.4174--5.8184--4.9194-
5 5 5 5 5 5
15.142 15.338 14.972 14,899 14.300 13.401
15.147± 15.339~_ 14.976414.898414.300± 13.4014-
2 8 1 2 1 2
69) 69)
~6°Dy 161Dy 16ZDy X~3Dy 164Dy
~< 8584~< 563zk ~< 859zk 51947704-
--6.9244--6.5464--6.1684--5.985zk --5.450±
5 5 5 5 5
15.406 15.028 14.650 14.467 13.932
15.415± 15.046414.652414.471 ± 13.9324-
2 7 1 1 1
164Er 166Er 167Er a68Er 17°Er
~ 501 4-125 ~< 580± 13 ~ 3894- 13 ~ 5804- 51 7504- 23
(15.744) 15.123 14.909 14.205 13.267
15.751 ± 1 15.128 zk 1 14.9044- 4 14.2094- 1 13.2624- 2
6~) 69) 69)
liZCd H3Cd 11*Cd llrCd
~ 56:k <~ 324~< 644~< 75=k
83 23 24 19 22
'.
(--7.2624-10) --6.6414- 5 --6.4274- 6 --5.7234- 6 --4.785:t- 5
69) 69) 69) 69)
69) 69)
69) 69) 69)
69) 69)
t6Syb 17°yb lVXYb 17ZYb 173Yb 174yb 176yb
3214532420745574263454544924-
16 19 10 11 8 11 10
--7.6474--6.861± --6.5994--6.1614--5.913=E --5.3594--4.2164-
7 6 5 5 5 5 5
16.129 15.343 15.081 14.643 14.395 13.841 12.698
16.0744-141 15.329± 17 15.0884- 13 14.6454- 8 14.3964- 4 13.8434- 4 12.695± 3
7o) zg) zg) z9) 67)
176Hf 177H.f 178[~.[f 179Hf lS°Hf
<~ 4354<~-3424464423045014-
34 14 15 15 25
--6.3974--6.0714--5.5314--5.2494--5.0114-
5 5 5 5 5
14.879 14.553 14.013 13.731 13.493
14.9264- 59 14.421 4- 42 14.0124- 5 13.7314- 3 13.4924- 3
zg) z9) z9) z9)
--5.7384- 5
14.220
14.2184- 37
7o)
(--6.2614-10) (--5.8104-10) --5.1244- 5 --4.4744- 5
(14.743) (14.292) 13.606 12.956
14.8954- 49 14.2434- 33 13.6064- 3 12.9544- 3
70) vo) 6s) 6s)
14.284 13.913 13.716 13.317
14.301-4- 10 13.9194- 3 13.715± 3 13.3184- 3
6s)
lSlTa
~, 354
182W 183W ls4W 186W
~ 5354- 32 ~ 164± 27 8234- 17 10504- 20
188Os
~ 6844~< 409477147784-
lS9Os 19°Os
192OS
26 18 19 16
--5.802± --5.4314--5.2344--4.8354-
5 5 5 5
2) Except for the Cd isotopes which are on a separate relative scale. b) Errors in keV. c) Errors are the same as for the Q-values.
67) 67)
67)
68) 68) 68)
224
M . A . OOTHOUDT AND N. M. HINTZ
spectrometer 16). Beam collimators were used at the entrance to the scattering chamber to give a 2 mm high by 1 mm wide beam spot on target. The (p, t) data were taken using at 3 ° (1.855 msr) entrance aperture to the spectrometer. A 2 ° (0.861 msr) and a ½° (0.211 msr) aperture were used for the (p, p) data. The Yb, Gd, Dy and Er targets were made by reducing the oxides with Ta powder in a resistively heated Ta boat and evaporating onto carbon foils 20 to 50/~gfcm z thick. The W and Ta targets were evaporated with an electron beam system. The Hf and Os targets were supplied by the Los Alamos Scientific Laboratory. Table 1 lists the composition of the isotopic materials as given by the supplier. No attempt was made to measure these compositions, but the strengths of peaks from contaminant isotopes were consistent with the listed compositions and the relative (p, t) cross sections measured with natural targets. For natural targets the natural abundances of the isotopes compiled by Fuller 17) were used. The isotopic targets contained 100 to 200 #g/cm 2 of the isotope of interest with the exception of a 440 pg/cm 2 171yb target and a 13/~g/cm 2 16Syb target. Absolute thicknesses for the isotopic H f and Yb targets were obtained using relative (p, t) cross sections from natural targets and (p, p) scattering on isotopic targets. No attempt was made to measure absolute thicknesses for most of the natural targets used. Tritons were collected using 50 #m Kodak NTB nuclear emulsion plates and position sensitive detectors (PSD)* in the focal plane of the spectrometer. Average resolution was 10 to 12 keV FWHM. Plates and detectors were calibrated by changing the magnetic field of the spectrometer to step a known peak (usually the ~76yb(p, t) ground state transition) across the detecting surface. Polyethylene foils 0.15/~m thick were used in front of the plates to stop singly charged e-particles observed at the same radius of curvature as the tritons. The plates were manually scanned in ¼ mm strips. The PSD's were 500 to 1000 pm thick with a nominal length of 3 cm and height of 1 cm. The 500 #m detectors stopped the highest energy tritons observed. In order to obtain complete excitation energy spectra, it was necessary to take runs at two or three different spectrometer magnetic fields to fill in the gaps between detectors. The signals from the PSD's were amplified and presented to the inputs of ADC's. Outputs from the ADC's were analyzed by an on-line computer using routines written a this laboratory as). Accurate ground State Q-values for the Yb(p, t) reactions were found by matching radii of curvature of the outgoing tritons. For the other isotopes studied, Q-values were found from measurements of relative excitation energies on plates. Further details may be found in ref. 8). The ground state Q-values and (p, t) cross sections for the lowestL = 0 transfers are given in table 2. The contents of table 2 are discussed in more detail in subsect. 4.4. t Supplied by Nuclear Diodes, Inc.
(p, t) R E A C T I O N S
225
3., Elastic and inelastic proton scattering To get absolute cross sections and proton optical-model parameters for the Yb(p, t) reactions, data on the lv6yb(p, p) reaction was taken. The 2 + and 4 + members of the ground state rotational band were also observed for 0~ab > 22½° and 35 ° respectively. Fig. 1 shows a typical spectrum. Carbon, oxygen and 4°Ca 176 YB (p,p) 176yB d
2+0
Ep =19.0 M e V
400 z < "r" (.3 5 0 0 r'~ ILl t3... 2 0 0 I"== 100
0+0
85°
0 L =
,4%
4+0
o
.......
0
...&, ............ I00
50
,J 200
150 CHANNEl
250
Fig. 1. Position sensitive detector s p e c t r u m for 176yb(p, p) to the first three m e m b e r s o f the g r o u n d state rotational band. Lines d r a w n to guide the eye.
I o_
b ~-
IzsYB ( p,p)lZSyB
o.2 0.1
5d
,
2O ~.
0+0 *
,
,
IO
03
"
"~
*
,
5
2+0
.
.El
°
E ~=
2 I
,~ b~
0.5
"Ol'o
0.2
°
.
~
....
".
t" ÷
;, {,,. ]
4+0
". °
I
0
20
~
,40
I
60
I
80
°
*
°
I
mo
°
°
•
*
I
~20
8cm
Fig. 2. A n g u l a r distributions for the first three m e m b e r s o f the g r o u n d state b a n d o f 17eyb excited by p r o t o n scattering. Values o f J~rK are indicated. Relative errors are s h o w n where they are larger thart t h e d a t a points. T h e absolute n o r m a l i z a t i o n error is a b o u t 10%. T h e solid line is the opticalm o d e l fit to the elastic scattering data u s i n g p a r a m e t e r set 5 o f table 3.
contaminants were identified by kinematics. Corrections were made for the presence of other Yb isotopes in the target by assuming the cross sections to members of ground bands for these "contaminants" are equal to those of 176yb" A partial justification for this assumption may be found in the 16 MeV proton scattering done by Kruse e t al. ~ 9) where it was found that the cross sections for 0 + and 2 + members of ground
226
M.A. OOTItOUDT AND N. M. HINTZ
bands were not greatly different for 172yb ' ~7 4 y b and 176yb" The angular distributions are plotted in fig. 2. Error bars are shown where they are larger t h a n the data points. The elastic scattering data was fit by the optical model [parameterized as in ref. zo)] with the automatic search routine R A R O M P zl). The C o u l o m b radius was taken f r o m a fit to electron scattering data 22) and spin-orbit parameters were taken f r o m the global parameters o f Becchetti and Greenlees 20). In order to avoid overTABLE3 Optical-model parameters for 176yb(p, p), E = 19 MeV Set
0 c) 1 2 3 4 5
Real
Imaginary
V (MeV)
r ") (fm)
a (fm)
57.824 56.4588 56.242 54.2057 55.6437 54.2725
1.17 1.202 1.2005 1.2306 1.2055 1.2315
0.75 0.665 0.642 0.6673 0.6587 0.6614
Wv (MeV)
WD (MeV)
r a) (fm)
Spin-orbit a (fm)
1.4811 9.5033 1 . 3 2 0.653 0.0623 10.8819 1.0614 1.0592 0.0006 10.6395 1.0929 1.110 11.0035 1.0754 1.0281 7.6155 1.4776 0.9509 12.5116 1.0253 1.0206
Z2 b)
V r a) a (MeV) (fro)(fm) 6.2 6.2 6.2 6.2
1.01 0.75 204 1.0 1.01 0.75 1.3 0.9 1.01 0.75 1.6 1.01 0.75 1.1
") Multiply by AT. ~) Chi-square per point, assuming a 3% error for all data points. c) Parameters from ref. 20). emphasizing the forward angles where the Y b first excited state and 4°Ca were n o t resolved f r o m the g r o u n d state, a standard error o f 3 % (approximately double the statistical error for most points) was used for all points. A normalization grid was performed to give the absolute cross section with an estimated 10 % error. Five parameter sets arrived at are shown in table 3 along with the Becchetti-Greenlees 20) global set. A t the conclusion o f the p r o t o n scattering run, the g r o u n d state transition in 176yb(p, t) was observed at 01,b = 27½ ° near the L = 0 maximum. Using the absolute normalization f r o m the p r o t o n scattering, the absolute cross section for the g r o u n d state transition is 492 pb/sr _ 10 %. D a t a for elastic p r o t o n scattering on 178Hf were taken f r o m 17½° t h r o u g h 27½ ° in 2½° steps. The cross section was assumed to have the same ratio to Rutherford as 176yb(p ' p). The absolute cross section for the 178Hf(p, t) ground state transition at 01,b = 27½ ° was f o u n d to be 464 ktb/sr + 20 %.
4. The (p, t) data and discussion Recent experimental a, 5 - 7 ) and theoretical 10, 12) investigations have shown that for deformed nuclei, the shapes o f (p, t) angular distributions for a given L-value are quite variable. (See also fig. 3,) D u e to the great length o f time necessary to
(P,T) ANGULAR
DISTRIBUTIONS L = 2 TRANSFERS
L=O TRANSFERS i l
t
t 174yB 0 keY : Q:-4.216 MeV
I
i
I
""
174yB, 76 keY Q=-4.292 MeV
q
] !
~,
I
172yB,78 t.',eV Q=-5.457 MeV
]~"N t
/ I
! \I/'
Q°-5.359MeV
I
,/ :' V: ~,174yB,1489 keY I
J\Mev I
E b
I
--~
b~
'~174yB,1886 keY ~-6.I02MeV
/ ~
I
4yB
'
i
i
168yB,87 keY Q=-6.948 MeV
I
~
i
~
1656 Re',
'
,
,
i
I
]
I I
I ~.L... , 168yB.984 key L "x...L.//I "% Q=-7.845 MeV 1174yB,2100 keV
.~
S
~
f X " Q=-6.3t6 MeV i
~
i
eV
,L°4! T,RANSFERS
t
i
I
"~@-.. ', ',
L l
t
I
>,,,"--T'~ , i 174yB,254 keY /I ~ ~"';,,,,_~Q=- 4.470 MeV
,
M
:
I I
' 168yB 0 keY
k
i
i
]
~
[
t
I
~, 168yB,284 keV Q=-ZI45 MeV
i
,
i
t
I
20
40
I
i
60 8c.m.
I
I
80
I
I
I
0
I0 0
[
20
I
40
60 ec.m.
80
I
I00
Fig. 3. Shapes for (!0, t) angular distributions labelled by excitation energy and Q-value. The cur are s m o o t h lines drawn t h r o u g h the data points. F o r each value of L, the curves are arranged vertic~ in order o f decreasing Q-value. These curves were used in C U R V E F I T as discussed in sect.
250O [L.
i
'
I
EXCITATION ENERGY (keV) 2000 1500 I000
'
174yB (p,f)172 YB Ep = 19MeV eL= 271/2°
~_ 400
200
500 0 6+ G.S. BANDo+ ] ~ I
2+ 0 +
Or)
")÷
¢o I00
2+ 0 ~~ f
t-0
-~r~
I/©
115 120 125 DISTANCE ALONG PLATE (cm)
150
Fig. 4. Triton spectrum for the t 7 4 y b ( p , t)172yb reaction. The K = 0 bands are indicated. C o n t a m i n a n t peaks are labelled " C " .
228
M.A. OOTHOUDT AND N. M. ItlNTZ
178HF, 0 keY O+OA
176HF, 0 keV O+OA
168yB,II98 keY 0 + 0 B
172yB, 1040 keY
168yB, 1340 keV (O+OC)
172yb,179l keV 52 keY O+OB
~90+Oc 168yB, 1543 keV O+OD
2 keV 178HF,1448 ReV
11~~'9
'~
2I " ~+yB,+°° 2504 keV
keY
/~ O+OB ,
3
f / ~ 178HF,1776 keV
~
0+0C
170yB, 0 NeV
E v
168yB, t904 key (O+OE)
(O+OE)
7B~
O+Oc
HF, 2024 keY O+OD
I HF, 1749 keV i O+OD
168YB, 2t60 heV (O+OF)
/1/ ~ !
174yB 2821keY
J
/
166yB,
/ 0 ,eV O+OA
IZa~F, 2316
keV
~(O+OEIl
I
I
1
0 20 40 60
i,
,~
¥~3, 2904 keV / ~ (04-0F)
I
I
I
0 20 40 60 0 20 40 60 Ocm
P //~ 166yB, 1045 keV (O+OB) f
0 20 40 60
Fig. 5. Angular distributions for 0 + states not shown in fig. 3. The residual nucleus and excitation energy are indicated. Curves are labelled by ]~K followed by a letter to distinguish bands. Lines are drawn to guide the eye. compute angular distributions with coupled-channel computer codes and to a lack of proper wave functions, L-values were identified by comparison of the data with the shapes of angular distributions for transfers to known states. Assignments made onthe basis of this phenomenological procedure are considered to be tentative unless
(p, t) REACTIONS
229
they agree with assignments made in other experiments or with I(I+ 1) spacings in proposed rotational bands. Identification of L-values by shapes of angular distributions was done with the computer code C U R V E F I T 23). A set of standard shapes (shown in fig. 3) for known L-transfers was read into the Williams Laboratory CDC 3100 computer. The code normalized the standard curves to an unknown curve to give the best chi-square per degree of freedom (X2), where the number of degrees of freedom equals the number of data points less the number of adjustable parameters. Here the only parameter is the normalization. The fits were ordered from best to worst Z: and displayed on an oscilloscope screen. Provision was also made for adding together two reference curves to obtain the best Zz. Angular distributions were taken on states to beyond 2.5 MeV of excitation energy in the H f a n d Yb isotopes (2.2 MeV in 166yb). A typicalspectrum is shown in fig. 4. For the Yb isotopes, searches were made up to 4 MeV of excitation at 27½° near the L = 0 maximum. No states were found beyond 2.5 MeV with strength greater than 5 ~ of the ground state transition. ( F o r ~66yb the search was to 3.3 MeV with no states stronger than 10 ~o of the ground state being found beyond 2.2 MeV.) Extensive angular distributions are shown in fig. 3 for states that have been assigned 0 + and for known 2 + and 4 + states. The curves are arranged vertically in order of Q-value. The L = 0 transfers have a very distinctive oscillatory pattern that is damped out as the Q-value becomes more negative. However, the shape is well defined by the cross sections at the laboratory angles indicated by the dashed vertical lines: 12½°, 27½°, 42½° and 55 °. The L = 2 shapes are quite variable, although the 7-vibration bandheads (third and fifth curves from the top) are similar to each other and the ground state band 2 + states (the remaining curves) are similar to each other. Only two extensive L = 4 angular distributions were taken. They do not resemble each other (especially at low angles) and the ~6syb 4 + is somewhat similar to the L = 2 distributions. Angular distributions for the other known L-transfers were not sufficiently characteristic to assign L-values. For most states data were taken only at the four angles identifying L = 0 transfers. These four angles will be called the "standard angles." The summed cross sections given in this paper are sums over these angles without a sin 0 factor. Unless otherwise noted, the percentage figures given to indicate the strength of excited states are the ratios of the summed cross sections for the state to the summed cross section for the lowest strong L = 0 state (the ground state for even-N targets). Angular distributions over the standard angles for 0 + states are shown in fig. 5. 4.1. SELECTION RULES The initial and final nuclear total angular momenta and parities are denoted Ji, Jr, zq and ~f. The total orbital and spin angular momenta of the transferred pair are designated L and S. Assuming the two neutrons in the triton are in an S -- 0 state with relative s-state motion [which has been calculated 24) to be 90 ~ of the
M. A. OOTHOUDT A N D N. M. HINTZ
230
TABLE4a States in 'TSHf Other experiments
This experiment J~rK (keV) 0 92 304 635 1088 1179 1272 1325 1387 1448 1510 1562 1643 1776 1816 1874 1947
(pb/sr)
0 10 10 10 10b) 10 10 10 10 c) I0 10 a) 10 10 10 10 a) 10 10
862.7 220.3 68.5 9.0 1.3 39.3 3.3 41.2 11.4 23.4 14.4 7.4 4.5 42.3 39.3 23.3 11.3
9.7 6.9 4.1 1.0 1.1 5.3 1.1 2.8 1.6 1.6 1.6 1.1 0.7 3.0 3.1 2.0 1.2
0+0A 2+0A 4+0A 6+0A 2+2
4+2 0+0B 2÷0B
0+0C (2+0C)
Ex (keY)
93 306 632
yTrK
Ref.
2+0 4+0 6+0
7s) 7s) 30)
1175 1269 1323 1384 1444 1515 1562 1636 1772 1819
2+2 3+2 3-2 4+2 0+0 2+(0) 2+(0) 4+0 0+0 2+0
3o) 3o) zo) 3o) 30) 3o) 30) 3o) 7s) 3o)
1954
(4)+(3)
30)
Other experiments
This experilnent Ex (keY) 1991 2024 2056 2121 2156 2203 2227 2286 2316 2371 2393 2435 2474 2572 2628 2668 2707
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
Xae.m. ~) ~b/sr) 11.1 42.3 10.1 14,5 10.5 20.8 23.9 19.2 44.2 42.0 18.5 14.2 16.8 6.8 6.3 14.1 6.9
1.2 2.1 1.2 1,6 1.4 2.0 2.1 2.1 3.1 3.2 2.1 1.6 1.5 1.3 1.1 2.0 1.3
.l n K
Ex (keV)
ar~rK
ReL
0+0D (2+0D)
(0+0E) (2+0E)
(2 +)
TABLE 4b States in xe6Hf Other experiments
This experiment ~K (keY) 0 87 287 600 1152 1231 1293 1314 1343 1362 1387 1511 1545 1607 1678
0 10 10 10 10 10 10 10 10 10 10 10 10 10 10
(/zb/sr) 812.7 15.3 219.2 8.8 44.1 4.3 8.3 1.5 95.0 4.6 28.7 3.0 63.8 4.6 32.0 3.1 55.5 4.3 6.7 2.5 19.2 2.5 5.6 1.1 15.4 1.9 3.7 2.1 15.3 1.8
0+0A 2+0A 4+0A 6+0A 0+0B 2+0B 0+0C 2+2 (2+0C)
E,~ (keV)
.17rK Ref.
88 290 597 1150 1227 1293 1313 1341
2+0 4+0 6+0 0+0 2+0 0+0 3-2 2+2
3s) as) as) as) 35) 35) 3~) 35)
1379
2+0
35)
1678
4+3
52)
(4+2)
Other experiments
This experiment Ex (keV) 1749 1796 1857 t953 2049 2069 2089 2136 2286 2304 2348 2389 2415 2448
10 10 10 10 10 10 10 10 10 10 10 a) 10 ~) 10 10
~ c . m . ") (~b/sr) 41.9 14.7 22.3 5.9 17.4 12.0 10.7 18.4 22.7 6.9 20.5 25.4 14.1 15.1
3.6 2.2 2.6 2.3 2.3 1.8 1.6 2.6 2.7 3.0 1.9 2.4 1.8 1.8
ar~K
0+0D (2+0D)
Ex (keY)
J~rK
1794 1862 t957 2049 2065
(3-) 2+ 2-2 1+ 1 2+(1)
2281 2307
(1-) 1+
Ref.
3s)
79) 36) a6) 36)
3s) 36)
TABLE 4C States in 174Yb This experiment Ex (keY)
0 76 254 526 895 1316 1386 1489 1520 1564 1608 1636 1676 1712 1783 1807 1852 1886 1959 2066 2100
~ . . . . ~) (/zb/sr)
0 5 5 5 5 5 5 5 5 5 10 5 10 5d) 5 5 5 ¢) 5 5 10~) 5
818.6 237.4 62.5 15.1 2.2 2.8 38.0 210.5 7.7 66.9 4.6 43.3 5.5 39.0 4.9 13.2 25,3 96.9 46.9 11.9 59.2
4.9 2.2 0.9 0,7 0.4 0.3 1.9 2.9 0.6 1.2 0.4 1.3 0.5 1.2 0.7 0.8 1.3 1.5 2.3 0.8 1.7
Other experiments JrrK
0+0A 2+0A 4+0A 6+0A 8+0A
,FrK
Ref.
76 253 526 892 1318 1380
2+0 4+0 6+0 8+0 2-2 3-2
47) 8o) so) 8o) 47) av)
1518 1559
(6 + ) 5-2
47) zv)
1630 1667 1702
2+2 6-2 3+2
37) 37) 37)
1800 1846 1884
4+2 3(5-)
3v) aT) 47)
Ex (keY)
0+0B 2+0B 2+2 4+0B 4+2 0+0C 2+0C o+oD
This experiment Ex (keV) 2127 2174 2246 2300 2342 2375 2436 2469 2520 2558 2588 2620 2662 2720 2753 2821 2840 2882 2904 3004 3042
5 5 d) 10 5 10 5~) 5 5 10 5 5 5 5 5 5 5 10 5 5 5 5
Other experiments
Y'~. . . . a) (ub/sr)
j~rK
13.4 0.8 55.1 2.4 9.0 1.0 4.8 0.6 12.3 0.9 7.3 1.3 12.7 0.9 19.0 1.1 6.2 0.9 50.2 2.4 31.6 1.4 10.5 1.2 29.6 1.6 11.9 1.2 9.0 1.2 12.0 1.5 21.3 1.6 7.6 1,0 14.5 1.3 24.2 1.6 6.7 1.4
4+0C 2+0D
4+0D
Ex (keV)
J~rK
Ref.
2184
2+2
av)
2333 2370
4+2 4+3
37)
a7)
(2 + )
(0+0E)
(0+0F)
TABLE 4d States in 172yb This experiment E~ (keY)
0 78 258 539 911 1040 1115 1154 1171 1218 1260 1281 1350 1410 1467 1508 1536 1556 1604 1628 1656 1705 1757 1791 1819
E ~ . . . . a) (/zb/sr)
0 886.3 11.1 5 287.6 6.3 5 80.1 2.3 5 15.9 1.1 I0 1.6 1.1 5 37.0 1.6 5 29.6 1.5 5 ¢) 14.1 1.0 5 8.6 0.9 5 18.2 1.2 5 c,~) 48.9 2.0 5 7.5 1.0 5 8.9 0.8 5 4.8 0.7 5 a) 68.1 2.2 5 10.2 1.0 10 6,7 0.7 10 7.5 0.8 5 7.0 0.9 10 4.1 0.7 5 d) 24.5 1.9 5 18.9 1.6 5 4.2 0.8 5 25.6 1.5 5 32.0 2.2
Other experiments d~rK
0+0A 2+0A 4+0A 6+0A 8+0A 0+0B 2+0B
4+0B
2+2A
2+2B 4+2A
0+0C
E~ (keV)
79 260 540 912 1043 1119 1155 1172 1222 1262 1283 1352 1405 1466 1510 1541 1549 1609 1632 1658 1708 1749 1794 1822
drrK
2+0 4+0 6+0 8+0 0+0 2+0 1-1 3+3 3-1 4+3 4+0 5-1 0+0 2+2 6+3 6-1 3+2 2+2 4+0 4+2 3-0 4+3 0+0 3-2
Ref.
a2) 8o) so) so) 32) a2) 48) s*) 51) 81) av) 51) a2) 81) 81) 51) 81) 8*) 48) 81) st) 81) az) 48)
This experiment E~ (keV) 1846 5 1892 5 1916 5 1954 5 2004 7 2025 7 2041 7 2060 7 ~'e) 2098 7 2184 7 2224 7 2254 12 2337 7e) 2364 7~) 2396 12 2436 7 2460 7 2483 7 c) 2540 10 ¢) 2580 10 2734 10 2786 10 2817 10 2832 10
Other experiments
Y,Oc.~. a) (/zb/sr)
,FrK
Ex (keV)
ylrK
Ref.
13.3 1.4 22.4 1.8 14.0 1.0 17.5 1.1 6.7 1.0 4.5 0.8 8.2 1.1 9.2 0.9 3.2 0.6 5.3 0.9 25.6 1.3 5.4 0.7 32.9 1.6 15.8 1.4 4.8 1.3 10.1 1.1 11.0 1.1 32.0 2.8 30.7 2.3 13.3 1.8 6.8 1.6 6.3 1.4 9,2 1.9 8.7 1.8
2+0C 0+0D
1850 1894
2+0 0+0
32) 32)
2+0D
1956 2009 2032 2047
2+0 1+t 32+1
z2) 81) 81) st)
2344
4+
81)
2480
(0) +
48)
2741 2786
9-5 8+8
sl) sl)
(2 +) (2 +)
(2 +) (0+0E)
232
M . A . OOTHOUDT A N D N. M. t t l N T Z TABLE 4e States in t 7 ° y b
This experiment E~ (keV)
Y'a¢.m. ") (~b/sr)
0 0 85 5 277 5 572 5 1072 5 1144 5 d)
904.2 323.9 77.0 11.3 15.6 72.2
9.8 6.6 3.1 1.2 1.4 2.3
1225 10 1331 5
8.1 24.4
1.0 1.8
1350 1360 1398 1478 1513 1534 1568 1602 1634 1657 1711
8.9 8.3 23.5 13.4 8.7 8.8 27.9 3.1 10.1 4.2 7.6
1.1 0.8 1.2 1.3 1.0 1.1 2.0 1.4 1.2 1.2 1.0
5 5 5 a) 5 5 5 5) 5 5 5 5 5
Other experiments J~K
E~
JrrK
Ref.
(keV) 0+0A 2+0A 4+0A 6+0A 0+0B 2+2, 2+0B 4+2, 4+0B
0+0C 2+0C
(4+0C)
This experiment Ex (keV)
84 278 573 1069 1138
2+0 4+0 6+0 0+0 2+2
as) s3) a7) a3) a3)
1780 1838 1871 1971 2001 2047
1222 1325
3+2 4+2
a7) 37)
2137 12 2171 7
1360 1400 1480 1512 1534 1566
1-0 30+0 1-0 2+0 0+0
3a) sT) 3a) aa) as) as)
1634 1651 1714
2+0 2+0 2+1
2229 2249 2281 2325 2352 2390 2437 2492 3a) 2539 54) 2595 54) 2678
5 c) 5 5 10 10 c) 7
7 7 7 7 12 7 12 7 7 7 7
~--,t~c.m.")
Other experiments
J~rK
(/~b/sr) 27.8 7.4 4.9 12.0 8.6 16.7
1.4 0.7 0.7 1.3 0.8 1.t
9.0 1.1 21.7 1.6 14.2 8.7 12.9 29.9 7.3 9.9 16.5 8.0 20.5 5.1 6.8
E~
J~rK
Ref.
(keV) 1783 1832
32+1
37) 5*)
2040
1+ 1
33)
2275
1-0
3s)
2533
1+0
3s)
(2 + )
1.2 0.9 1.2 2.3 1.3 1.7 1.4 1.4 2.1 1.6 1.8
TABLE 4f States in 16Syb This experiment
.E,~
Y'ac.m. ~)
(keV)
(#b/sr)
0 87 284 586 886 943 984 1064 1098 1161 1168 1198 1231 1277 1340 1480 1543 1600
0 5 5 5 10b) l0 b) 5 5 5 7e) 5 5) 5 5 5 7") 5 c) 5 5a)
938.1 361.9 93.2 10.8 2.9 1.9 45.8 4.3 5.6 3.1 14.4 19.7 7.8 20.1 2.8 24.2 26.5 47.0
11.4 6.9 3.9 0.7 1.1 1.1 1.7 0.5 0.8 1.1 1.0 1.2 0.5 1.6 0.6 1.2 1.9 2.3
Other experiments
JrrK
E,,
JrrK
Ref.
(keY) 0+0A 2+0A 4+0A 6+0A
2+2
4+2 0+0B 2+0B (0+0C) (4+0B) 0+0D (2+0D)
88 287 585
2+0 4+0 6+0
s*) 34) 34)
984 1067
2+2 3+2
s4) a*)
1156 1171 1197 1234
0+0 4+2 0+0 2+0
34) 34) 82) s4)
1475 1543 1595
30+0 3-
37) s2) 37)
This experiment E~ (keV) 1698 1727 1767 1793 1859 1904 1973 1993 2092 2121 2160 2173 2186 2292 2327 2360 2464 2500
5 5 5 5 5 °) 5 5 5 7 7 7 12 7 7 7 12 12 12
Z(rc.m" a) (#b/sO 10.7 5.7 6.2 6.7 16.7 37.4 9.0 30.1 5.8 5.2 12.2 6.4 7.3 7.7 6.9 6.4 8.8 4.7
0.9 0.7 0.7 0.6 1.1 1.7 1.1 2.0 1.1 1.2 1.2 1.5 1.2 1.1 1.0 1.2 1.5 1.5
Other experiments
jz, K
Ex
JZrK
Ref.
(keV)
(4+0D) 1770
(0+0E) (2+0E)
(0+0F)
5-
a*)
(p, t) REACTIONS
233
TABLE 4g States in 166yb Other experiments
This experiment Ex (keV) 0 0 101 10 329 10 a) b) ~) ~) D
~r~.m. a) (#b/sr)
yzrK
646.9 22.0 267.2 16.1 54.8 8.0
0+0A 2+0A 4+0A
Ex (keV)
102 330
JrrK
2+0 4+0
Ref.
s3) 8a)
This experiment Ex (keV) 931 10 1043 10 1581 10
Z~r.... a) ~b/sr) 57.7 7.8 75.9 11.0 29.8 15.5
Other experiments JZrK
Ex (keV)
Jr:K
2+
(0+0B)
Sum of center-of-mass cross sections over standard angles. Unidentified contaminant? Possible doublet. Doublet. Contaminant subtracted.
time], it has been shown 2 5) that the following approximate selection rules hold:
IJi-Jel < L < Ji+Jf, = ( - 1 ) L.
(1)
(2)
For deformed nuclei further selection rules may be found. Let K i and Ke be the projections on the symmetry (z) axis of the total initial and final angular momentum in the intrinsic frame. Sufficient angular momentum must be supplied to satisfy eq. (1) and to change the projection on the z-axis; thus t __> [gf-K~[.
(3)
The Z and A selection rules result from consideration of the projections on the z-axis of the spin and orbital angular momenta of the transferred neutrons 26). The neutrons are in Nilsson levels labelled by the asymptotic quantum numbers £2[NnA]S, where f2, A and 27 are the projections of the total, orbital and spin angular momenta of the orbital on the z-axis, N is the total number of harmonic oscillator quanta and n is the number of quanta on the z-axis. The parity of the orbital is ( - 1)N. A pair of particles may be picked up with the projections of their individual total angular momenta parallel ( + sign) or antiparallel ( - sign):
01 [N1 n, A1 ]271+-f2z [N2nz A2 ]S=.
(4)
The projection of the total angular momentum of the pair on the z-axis is K = 12, +122 . If we assume as before S = 0, then we must have 271+2:2 = ~ -
& = 0.
(5)
Further we note that L must be large enough to give the required change in the A components, i.e., Z > [hl+_hzl, (6)
Ref.
234
M . A . O O T H O U D T A N D N. M. H I N T Z
where the sign is chosen in agreement with eq. (4). Eqs. (5) and (6) will be referred to jointly as the 2J-A rule.
I80H F(p,f) 178HF (0 +)
0+
0+
I1
I OOC
(5+)
Fp
0+
0 +
I
I-12~
o+111
I
I
I
:::k
0+
~z2-
(2 + )
I00
I
v
I
v
b k'q
i
,
i
j
i
i
i
2.0
2.5
1.5
1.0
0,5
0
EX (MeV)
Fig. 6. Schematic spectrum for 18 o E f ( p ' t)17 SHf" F o r each state the s u m m e d cross section is plotted at the p r o p e r excitation energy. The s u m is over standards angles (01.b = 12½°, 27½°, 42½° and 55°). K n o w n rotational bands are indicated. Values o f K ~r are s h o w n for bands and J~ for some individual states. The descending vertical lines are longer if experimental candidates for the state exist. I
I
]
]
i
J
L
I
i
I'
4
I
]
J
I
~
I
J
i
J
i
178 HF(p,f)176HF
I000
--
0+ 0 ÷
0+
I~
I-] I-]
] I
(6+)
5+
,~?~l~l {+ I
--. I O0
i + I+
gb b-q
0+
2+
' 12-
~ ]
~q~ Io -
2.5
2.0
L5
1.0
0,5
0
Ex (MeV)
Fig. 7. Schematic spectrum for 17 s H f ( p ' t)~v6I_If. See caption to fig. 6.
Aside from the assumption that the neutrons in the triton are in an S = 0 state, t h e above selection rules are based on the exactness of the asymptotic quantum
(p, t) REACTIONS
235
numbers for finite ft. Rassey 27) has shown that for a finite deformation the wave function is a linear combination of asymptotic states. In the rare earth region the '
i
,
I
i
I
t
i
l
,
i
,
,
I
,
i
i
i
I
~
i
i
i
i
i
176 YB(p,t) 174yB (o +)
0+
0+
0 4-
O+ [---]-] 3- I---~
-~ I000 ..Q
2+ 5+[--~
::L
i
F-P
I00
z-
q-T]
Q:>
b
Pq
I0
Jlll,l ,II[I lI I
1 I
2.5
t
i
t
,
I
i
2.0
I
J
i
i
i
1.5
i I i i i
t,0
f t l ,
0.5
0
Ex (MeV)
Fig. 8. Schematic spectrum for 176yb(p, t)tV4Yb. See caption to fig. 6.
I74yB(p,t) 172yB 0+
FTTq
_Q
(0+) I00
q~
b I0
0 4-
0+
0+
~,- I 0 0 0
::L
2+
I+
I~ 3-
l
3-2 +
5+
I
Ilti ,:iliil,]iill I
2.5
2.0
t.5
1.0
0.5
0
Ex (MeV)
Fig. 9. Schematic spectrum for 17~yb(p, t)t72yb. See caption to fig. 6.
asymptotic label has the largest amplitude, although there are significant c o m p o n e n t s which allow relaxation of the Z - A rule. Thus 27 = 1 two-quasiparticle states m a y be p o p u l a t e d in (p, t), but m o r e weakly than I; = 0 states.
M. A. O O T H O U D T
236
A N D N. M. H I N T Z
4.2. E V E N - N Y b A N D I-If T A R G E T S
D a t a for the states observed in this experiment are given in table 4. Excitation energies and errors (keV) are given in the first two columns. The next two columns J
'
'
J
'
'
'
i
'
'
'
'
I
. . . .
J
....
' ' '
172yB(p,t) 170yB 0+ I000
0+
0 +
J
I+ I+ 0 +
FTTi ( 2 +)
I
:::L IOO
b
3"
3-
I
I
llla,JI,I,1111 I
t0
J
2.5
J
2.0
i
1.0
15
0.5
Ex (MeV)
Fig. 10. Schematic s p e c t r u m for 172yb(P, t)17°Yb. See caption to fig. 6.
170yB (p,t) 168yB (O~i (0 +)
0 +
O+
0 +
A 03
"~ 1o o o _Q ::L
•
I
'"
3- 3-(0+) I L
III
v
I00 b I0
i
2.5
20 .
1.5 Ex (MeV)
1,0
0.5
o
Fig. 11. Schematic s p e c t r u m for lV°Yb(p, t)168Yb. See caption to fig. 6.
give the sum of the cross section over standard angles and its relative error. Absolute cross section errors are approximately 10 % for Yb and 20 % for H f isotopes. As-
(p, t) REACTIONS
237
168yB(p,t)166yB (0 +)
0+
2+
I000
1
n :::L I00 b tO
t
,
i
2.5
J
[
1
~
,
i
2.0
[
B
i
i
i i i i i
i
1.5 I0 Ex (MeV)
i
0.5
0
Fig. 12. Schematic spectrum for 168yb(p, t)166yb. See caption to fig. 6. Because of the poor quality of the 168Yb target, only the strongest states could be identified.
1ooo -
1
I
dodw ~b/sr
I
i
1
I
ybt76(p,t) Yb I~ ~
I00
Ep = 19 MeV
~
O÷
2+ I0 4+
•~ 0o
F
20 °
t
40 °
I
t
60 °
80 °
I
I00 o
8CM
Fig. 13. Angular distributions for members of the ground state band of 174yb populated by 176yb(p, t). Lines are drawn to guide the eye.
238
M . A . O O T I - I O U D T A N D N. M. H I N T Z
signments for J~K made in this experiment are given in the next column. A letter after the K assignment indicates which of two or more bands with the same K-value the state belongs to. The final columns give a partial listing of excitation energies
.,oool I
/
GROUND
STATE
BANDS
"
+
P,,l
-YB AHF
I.o ~-
•
I
2+ 0.1
+
o b Pq A
4+ i
0.01
¢
-.) v
b P4 0.001
I 96
i 98
RESIDUAL
t
f I00
I02
NEUTRON
4'
•
t'
8+
i 104
r 106
6+
NUMBER
Fig. 14. S u m m e d cross sections for g r o u n d states a n d ratio to the g r o u n d state s u m for m e m b e r s o f g r o u n d state bands. T h e s u m is over s t a n d a r d angles. T h e g r o u n d state s u m s h a v e a n absolute e r r o r o f 10% a n d 2070 for Y b a n d H f respectively. Solid lines indicate values o f ( A / G ) 2 normalized to t h e data. T h e s a m e n o r m a l i z a t i o n is u s e d for I-If a n d Yb.
I "-~" ;
- 5{-
~,~2 I"- ~ 2 . 0 ~ ~_.-[ - " t Sf0 ~ t
x,,~
K=O BANDHEADS 4% 2"1"2"1"
5" O i
i.0r
5% - - 5 % I' 5%-- " 5*/. I°z~3°1:~6./~ 26% -I ~ , 5 */* 0~/~ o ~ - - - - ~ v . . _ * --,04%8% . t~-5o/< I2% <0-~*2% -c--'2°/o 4% - - 12Y. . . . , I%
7%,2%
,
~
5%
--
5%
--
Fig. 15. The E = 0 bandheads in the Y b and H £ isotopes studied. Percentages give ratios of summed, cross sections for excited 0 + states to the ground state. Dashed lines indicate 0 + states not seen in this experiment; u p p e r limits on the strengths o f these states in (p, t) are indicated.
(keV) and assignments from other experiments. "Contaminant subtracted" means that a state from anothe r isotope of the element considered has been subtracted to give the cross section indicated. In figs. 6-12 we plot the summed cross sections
(p, t) REACTIONS
239
versus excitation energy. Known bands are indicated above the schematic spectra; short lines correspond to known states not observed in (p, t). 4.2.1. Ground state K = 0 bands. Data were taken on ground state rotational bands up to the 6 + member in most cases and to the 8 + member in some. Fig. 13 shows extensive angular distributions for members of the ground state rotational band up to the 8 + member populated by 176yb(p, t)174yb. Fig. 14 shows ground state strengths and the ratios of summed cross sections in the ground band to the ground state strength. In the PWBA-BCS approximation the ground state cross section is proportional to (A[G) 2, where A is the BCS gap parameter and G is the strength of the pairing force 2s). This term is plotted in fig. 14 using G = 20/A MeV. Here and later A is set equal to the odd-even mass differences of Whineray et al. 29). Average values of G and A over the target and residual nucleus are plotted, since it is not clear which should be used. The fit is poor, reflecting the inadequacy of the PWBA. The sharp drop for the 4 + and 6 + cross sections of 176Hf with respect to 1 7 4 y b may be due to an abrupt change in the Y4o moments. From inelastic e-scattering 15), /?4 is known to change from -0.054 for 176yb to -0.072 for 178Hf. 4.2.2. Other K = 0 bands. As previously reported 2), a number of excited 0 + states are populated by the (p, t) reaction in the Yb and Hf isotopes. Angular distributions for the 0 + states are plotted in figs. 3 and 5. The proposed 0 + states are shown in fig, 15 along with 0 + states not seen in this experiment but reported by other workers 30- 34). The 0 + states were generally observed with 2 to 5 ~ of the ground state summed cross section. The population of 0 + states in 176Hf at 1152 keV (with 12 ~ of the ground state strength) and 1293 keV (8 ~), in 174yb at 1489 keV (26 9/o), 1886 keV (12 ~o), and 2100 keV (7 ~o) and in 166yb at 1043 keV (12 9/0) is therefore distinctly anomalous. The 0 + states in 176Hf have been previously observed 35, 36) in the decay of 176Ta" The strong Yb 0 + states have not been previously reported, but the excellent L = 0 shapes for the angular distributions (figs. 3 and 5) and the presence of well developed rotational bands built on these states for 174yb (table 4c) make 0 + assignments very certain. For ~66yb background was too high to observe higher band members for the 1043 state. The suggested 2 + (2174 keV) and 4 + (2342 keV) members of the 2100 keV band in ~74yb are near members of a K = 2 band observed by Burke and Elbek 37) with a 2+2 at 2184 keV and 4+2 at 2333 keV. The 2174 keV peak is very broad in the (p, t) spectrum indicating it is an unresolved doublet; it has a reasonably good L = 2 shape indicating both components of the doublet are 2 +. Several explanations have been put forward to account for the numerous 0 + states observed in the rare earth region: I (i) The /?-vibrations considered by B~s 3s) ~esult from residual quadrupole interactions which are typically written i i
-- z f ( r , , rz)Y*,(l!')Y2~(2).
240
M.A. OOTHOUDT
A N D N . M. H I N T Z
SINGLE NEUTRON
LEVELS
3[ 501] . . . . . . . . . . . . . . . . . . . . . . . . . . . /-----
[1[6153 . . . . . . . . . . . . . . . . . . . . . , i't503]
,, -71514.]-.
1,,,--
;z:
Y.
~,
/ .~.__71503] I ~. . . . 5[5 t2]t
.....
T
,~------7'[st4]j
--/" q---~l"-" ~-~- 5ts~q
---~,".--'~/
",~.~,L
,~,o~-:-'-~,':--,"~/
,=,, ,[62,,,. . . .
/
." ,
/'::'~*- '~"4
\
,
{;;
"\'-~-- " ~ ' f - I - " , ' - 4 N J .... ~T~-, I
1'/
/'ki,._l_
;"
t
t 31501] -f 1116151~
/ .........
I
315e~;q
..sE523J/
3 [6513----~----~-e-" . - - - - - - - - - - - - - - - - - - - - - - - - ] -°-t~"/ I I [ 5 0 5 - - : : i ~ / - ' : - e - e - - - - - 4 ~ - - - - -e-~---- -e-o- --- -,~4~ ----~,-oI [ 660]'/~
I [660]~ 111502] I
166yBlTOyB 172yBI74yB I7~6yB178HFf8°H F
Fig. 16. U n p e r t u r b e d s i n g l e - n e u t r o n levels obtained by ref. 46). T h e asymptotic q u a n t u m n u m b e r s (2~2)[Nn=A]are indicated. T h e levels s h o w n are for t h e o d d - N n u c l e u s with one less n e u t r o n t h a n t h e nucleus indicated. Circles indicate levels filled in t h e limit o f zero residual interaction. Vertical bars are twice t h e odd-even m a s s difference.
K=O
EXCITED
BANDS
-YB AHF
LI.I 1.0 .6 z r'n .2
,B
,O
•O
2 +
ACE)
,B,D "G IB
B,(C) • (D)
,[E)
0 Io
.6
.2
4 +
-CO) • (D)
,B
,B "0
r
i
.I I
i
/
98 I00 102 104 106 RESIDUAL NEUTRON NUMBER
96
Fig. 17. Ratios o f t h e s u m m e d cross sections for J = 2 a n d 4 states in excited K -- 0 b a n d s to t h e s u m m e d cross section for t h e b a n d h e a d . Letters to the right o f the points indicate b a n d m e m b e r s h i p (see table 4). States w h i c h are doublets or are n o t resolved f r o m k n o w n 2+2 a n d 3 - states are n o t shown.
(p, t) REACTIONS
241
where X is the strength of the interaction, r~ is the vector to particle i a n d f ( r 1 , r2) is some function of the radial coordinates. These collective excitations are expected to appear in the energy gap. In general they are not expected to have more than a few percent of the ground state strength in two-neutron transfer reactions 14). (ii) Spin-quadrupole excitations a 9) are due to interactions of the type
--tcf(rl, r2)(~1 Y2(1))~(tr2 Y2(2))2,, where a~ is a Pauli spin matrix. Such an interaction can also result in 0 + states within the energy gap. It is expected 40) that the spin-quadrupole force will reduce the (p, t) strength to states of type (i). (iii) Pairing vibrations are due to the residual pairing interaction and should appear above the energy gap. The strength of the two-nucleon transfer reaction to such excited states is normally expected to be only a few percent of the ground state strength in the rare earth region 41). (iv) A combination of the monopole and quadrupole pairing interactions acting on particles moving in Nilsson single-particle levels with a nonuniform distribution of prolate and oblate orbitals around the Fermi surface has been shown 14, 42) to give excited 0 + states with (p, t) cross sections on the order of 10 to 20 ~ of the ground state strength in the actinide region. The cross sections are stable against changes in the neutron number. (v) In regions of changing equilibrium deformation the (p, t) reaction may populate a deformed 0 + excited state strongly in a nucleus with a spherical ground state if the target nucleus is deformed. Such states have been seen in Nd [ref. 43)], Sm [ref. 1)] and Gd [refs. 4, 6)] near N (neutron number) = 88. The 12 ~ state in 166yb (N = 96) fits in well with the systematics observed by Maher et aL 5), Else et al. 6) and Fleming et al. 4). They observe Gd, Dy and Er isotopes over the range 90 _< N < 96 which have excited 0 + states populated by (p, t) with 10 to 20 ~ of the ground state strength. The fourth mechanism described above might explain such systematics 14). The very strong states in 174yb and 176Hf (both N = 104) do not follow such a systematic trend. Nor do the other mechanisms for strong 0 + states apply since the H f and Yb nuclei are well deformed and have an approximately uniform distribution of oblate and prolate orbitals. These states might be explained 2) in terms of the pairing phase transition described by B~s and Broglia 41), Broglia et al. 44) and B~s et al. 45). In these papers it is shown that if there is a gap in the single-particle spectrum near the Fermi surface on the order of twice the energy gap parameter A and if there is sufficient degeneracy above and below the gap, appreciable strength will appear in the L = 0 two-neutron transfer reaction. The gap in the single-particle spectrum may be said to cause a phase transition from superconducting to normal. In fig. 16 the single-neutron levels determined by Ogle et al. 46) from the experimental energies of one-quasiparticle states are shown for several Yb and H f isotopes. The vertical bars indicate values of 2A. The condition for strong excited 0 + states is
242
M.A. OOTHOUDT AND N. M. ItINTZ
best met for 176yb and 178Hf targets. Using the parameters shown for 176yb(p, t) and calculations for a simplified two-level model 45) with a pair degeneracy of five per level, we find as a crude estimate o-~x/O-g.~"~ 0.67, whereas experiment gives 0.26. Large excited state cross sections are also predicted for some of the other Yb nuclei. The discrepancy between this model and experiment is not surprising, since the two-level model is certainly an oversimplification of the nuclear structure. Reaction effects have also been ignored (see sect. 5). Thus while we have a plausible mechanism for the strong population of 0 + states by (p, t) that applies to the Yb and H f isotopes, we do not have a detailed quantitative explanation. In fig. 17 the ratios of cross sections of the 2 + and 4 + members of excited K -- 0 bands to their bandheads are shown. Several states not resolved from known members c f gamma and octupole bands are not shown. A comparison of figs. 14 and 17 shows that the ratios for the excited bands are a factor of 1.5 to 2 times the ratios for ground bands. This increase in ratios might be due to mixing with nearby v-vibrational bands. A number of previously reported 0 + states were either not observed in this experiment or were observed With angular distributions different from the normal L = 0 shape. In the former class is the 1199 keV 0 + state 30) in 178Hf. A state is observed at 1272 keV near the 2 + member of this band, but in view of the low population of the bandhead (upper limit of 0.5 ~ of the ground state strength) the 1272 state is more likely to: be the 3 + member of the ;~-vibrational band. The 1434 keV 0 + state 3 o) is not seen either, although it is possible that this state was not resolved from the 1444 keV 0 + and their sum is the 1448 keV state reported here. However, the peak shape in the (p, t) spectrum does not indicate a doublet. In 174yb a 0 + state was reported 31) at 1305 keV in 7-ray work using a NaI detector. The (p, t) spectrum was carefully searched and an upper limit of 0 . 4 ~ of the ground state strength was placed on this state. If, as suggested by Wilson and Pool 31), the 1228 keV 7-ray depopulating this state corresponds to the 1245 keV 7-ray reported by other workers 47), the state reported as a 1305 keV 0 + is actually the 1316 keV 2-2. The 1410 and 1628 keV peaks observed in 17Zyb are near the 0 + and 4 + members of a K = 0 band at 1405 and 1632 keV reported in (n, ~) work 32, 4s). (The 2+0 at 1476 keV would not have been resolved from the strong 7-vibrational bandhead.) T h e angular distribution for the 1410 state does not have the oscillatory L = 0 character, although it is possible that for such a weak state (less than 0.5 ~ of the ground state strength) the minima could be filled in by the normally insignificant triton background. Peaks at 1225 and 1568 keV in ! 7Oyb are near 0 + states reported at 1228 and 1566 keV in 7-decay work 33). Again these states do not have an L = 0 character. The 1568 keV peak is quite strong (3 ~o of the ground state strength) so that an L = 0 character should have been observed. A peak at 1634 keV near the 2 + member of the 1568 keV band was also observed with about 30 ~ of the 1568 keV state strength. Without
(p, t) R E A C T I O N S
243
convincing evidence for a 0 + at 1568 keV, we cannot assign this peak as a 2 + on the basis of the shape of its angular distribution. I n . l e a y b peaks at 1161 and 1231 keV are near 0+0 (1156 keV) and 2+0 (1234 keY) states reported in ?-ray and conversion electron work a4). The 1161 keV peak does not have an L = 0 character, although for such a weak state background might
GAMMA VIBRATIONAl BANDS 16
"',
", 14
GROUND STATE BANDS /
G A M M A ~ "- - " ~ -
BANDS
"Fv
~
12 i
>-
1.5
l
i
1
i
i
.YB •
"-.
o Ld
~od '-? bJ O2
4
m~
2 •
f
i
i
i
I
i
r
i
~
~
t
60
U3
20 ~0
.~].~ b~
6
2 , 9c8 I00 , , ' 96 102 104 I'06 RESIDUAL NEUTRON NUMBER
Fig. 18. Properties of 7-vibrational bands. Inertial parameters, excitation energies, B(E2) values 37.65) and summed (p, t) cross section are shown.
fill in the minima. The 1231 keV peak is twice as strong as the 1161 keV peak (unlike other 2+0 states which are at most equal in strength to their bandheads) and does not have an L = 2 shape. 4.2.3. ?-vibrational bands. Bands with positive parity and K = 2 may be interpreted as quadrupole vibrations of the nuclear surface, or microscopically as superpositions of two-quasiparticle states 49). States were observed at the energies of previously reported ?-vibration bandheads with 5 to 10 % of the ground state strength. Angular distributions for these states (see fig. 3, 1636 keV in l~4Yb and 984 keV in 168yb ) are more oscillatory than those for the 2 + members of the ground state band. In spite of the general instability of shapes for angular distributions, we have as-
244
M.A. OOTHOUDT AND N. M. HINTZ
signed these states to be the 2 + bandheads for y-vibrations on the basis of their excitation energies. A candidate for the previously unobserved y-vibration bandhead in 166yb has been found at 931 keV. The 4 + and, in some cases, the 3 + members o f these bands were identified by comparison with the known excitation energies of these states. The 4+2 of ~76Hf is assigned to the state at 1545 keV using the ground state band moment of inertia and the I(I+ 1) energy spacing rule. (The use of the ground state band moment of inertia is justified in fig. 18.) States of unnatural parity such as 3 + are not expected to be excited except very weakly by the (p, t) reaction (see subsect. 4.1). The excitation mechanism may be a two-step process such as (p, t) to the 2 + bandhead followed by inelastic excitation to the 3 + by the exiting triton or perhaps inelastic excitation of the target by the incoming proton to the 2 + of the ground state band followed by an L = 2 (p, t) transfer to the 3 +. In fig. 18 cross sections and other properties of these bands are shown. The cross sections for the ~7°yb 2+2 and 4+2 are upper limits only, since they are not resolved from the 2+0 and 4+0 members of the 1072 keV K = 0 band. For the H f and heavier Yb nuclei, B(E2) values are low and excitation energies high, indicating a low degree of collectivity for these states. The (p, t) cross sections do not follow these trends. In addition to these K = 2 bands, a number of other possible 2 + states were identified by the shapes of their angular distributions with up to three-quarters of the yvibration bandhead strength (see table 4). Since no 0 + states were observed below them with a reasonable energy spacing, they are assumed to be J~K = 2+2. These states could also be 2+1, but angular distributions for known 2+1 states do not resemble either y-vibrational or ground state 2 +. The 1604 keV state in ~V2Yb is assigned by Burke a7) as a K = 2, {~[512] -½1521]} two-quasiparticle state, but by the Z-A selection rule such a configuration should not be observed in (p, t). Thus the strength of the 1604 state is either a measure of the mixing of other configurations into this state (e.g., ½[521] + ~ [521]) or a measure of the breakdown of the Z-A rule due to the finite value of fl as discussed in subsect. 4.1. 4.2.4. Octupole vibrations. Several peaks were observed near the energies of previously identified octupole states. The shapes of angular distributions were not distinctive enough to assign L-values. Assignments are therefore made only on the basis of excitation energy. Theoretical calculations by Neerg~rd and Vogel 5 o) show that these are not very collective octupole states, since most of the E3 strength is expected to be at very high energy; only a small fraction is brought down below 2 MeV by the spin-orbit part of the nuclear potential. They also predict that for Yb and H f nuclei, the lowest octupole states will be K = 1 or 2. Due to strong Coriolis mixing, however, the K quantum number is not expected to be very good s o). In fig. 19 excitation energies and summed cross sections are shown for 3- states observed. The K-assignments are shown where known from other experiments. The 3 - states are excited with 2 to 5 ~ of the ground state strength. In 172yb members of a K = 1 band are seen with J~K, excitation energy and sum-
(p, t) R E A C T I O N S
245
med cross section of 1-1, 1154 keV, 14 #b/sr; 3-1, 1218 keV, 18 #b/sr; and 5-1, 1350 keV, 9 #b/sr. It is also possible that the 1536 and 1556 keV states are 6- and 7members of this band 51). The 1556 keV state is, however, near the 3+2A at 1549 keV. The alternative assignment of O'Neil and Burke 51) of 7- to the 1536 keV state seems more likely, since a 6- with unnatural parity and high L-value would not be expected to be populated with much strength. In 174yb the 2- (1316 keV) and 3(1386 keV) members of a K = 2 band are seen with summed cross sections of 3 and 38 #b/sr respectively. Burke and Elbek 37) find an inertial parameter h2/2J = 10.0 keV for this band and predict the 5-2 at 1564 keV near the 2+0B. Excitation energies
3" S T A T E S 2.0 A
AHF e2
• YB
>
•
*0 v
>..
1.5 •
Or" L.d
.2 A2
A2
ol
Z laJ
1.0 I
f
I
•
•
!
I
I
I
I
62 ~,2 •
A2
•2
I
I
"~ so "~
zL IE
40
20
b~
,o
io 6 4 I
I
96 98 I00 102 RESIDUAL NEUTRON
104 106 NUMBER
Fig. 19. S u m m e d cross sections a n d excitation energies for 3 - states. N u m b e r s to the right o f the p o i n t s give K q u a n t u m n u m b e r s . T h e 16Syb points are u p p e r limits only, since they are n o t resolved f r o m the 4+0B a n d 2 + 0 D states.
from this experiment give h z / 2 J = 11.7 keV and the 5-2 at about 1596 keV. Near this energy (1608 keV) we find a weak peak (summed cross section of 4.6 #b/sr) which may be the 5-2. However, due to the strong Coriolis mixing predicted 5 o) for octupole bands, it is not clear whether a simple I(I+ 1) rule is sufficient to predict excitation energies. Several other 1- states are indicated in table 4. 4.3. O D D - N Y b T A R G E T S
The study of the odd-N 173yb and 171yb targets with the (p, t) reaction was more difficult than the study of the even-N targets. Experimentally, the high density of weak states populated at low excitation energy in the odd-N nuclei (see figs. 20 and 21) made resolution of peaks difficult. Interpretation of the spectra was als0 more diffi-
246
M . A . OOTHOUDT AND N. M. HINTZ
cult, since the total angular m o m e n t u m of the final state is not uniquely determined by the L-value of the transfer when the target angular m o m e n t u m is not zero. For ~
,
,
~
~
J
J
i,
i
i
i
~
,
I
'
d
;
E
j
,
,
J
173yB(p,t) 17tyB tE[5os] !. I
]
x_
I000
~"
I
i
5 ~BAND
,
~
=
,
i
J
[
11521]
'
'
7 [6s31 i
5 [512]
5[~i~ I
: i :J
(I-) BAND 5[72SI]
I00
~
, ; ]i
1,4 I0
,
~
r
J
:
J
2.5
, I
L
2.0
~
i
I
t
1.5 ( MeV )
Ex
]1
,
i
,
1.0
I 0.5
0
Fig. 20. Schematic spectrum for 17Syb(p, t)171yb. For each state the summed cross section is plotted at the proper excitation energy. The sum is over standard angles. The major single-particle components o f rotational bands are indicated by giving the asymptotic quantum numbers (2#2) [Nm_A]. The " 5 - band" and " ( 1 - ) band" are possible bands built on vibrations of the even-N core. i
i
i
171YB ( p , t ) 169yB 7 E6ss] 5 [5~2] I
I000
I
I
L
-i
o~ t'b
I [510]
IO0--
s[521]
4 ! t
b %1
tO:L_ t-
i
2.5
i
J
2.0
i
i
t
t
I
L
1.5
~
,
~
i
1.0
I
,11 i
i
0.5
Ex ( M e V )
Fig. 21. Schematic spectrum for t71yb(p, t)169yb. See caption to fig. 20. Possible ½-- states populated by L = 0 transfers are indicated by (1).
this reason only dominant L-values assigned with C U R V E F I T are given for odd-N nuclei in this experiment. In general, we assume that the (p, t) cross section decreases rapidly as L increases as for the even-N nuclei. Thus we assume if a member of a
(p, t) REACTIONS
247
band allowed by selection rules is not seen, higher band members will be too weak to be seen. However, the operation of the A selection rule can modify this assumption. A further problem for the 173yb target is that with J~ = ~ - it is usually possible for the reaction to proceed by more than one L-value. (For 171yb(p ' t) with ground state J~ = ½-, parity and angular m o m e n t u m conservation allow only one L-value.) For example, the strong " L = 0 " transfer to the ~[512] state at 122 keV in 171yb may go by L = 0, 2 or 4. An extensive angular distribution was taken on the 122 keV state to test the contribution of L-values other than 0. An option in C U R V E F I T was used which adjusts the sum of two standard curves to the data to give the best Z 2 per point. The best fit (Z z = 4) was obtained from the sum of the 0 + and 2 + members of the 1 7 4 y b ground band. The i 7 4 y b ground state L = 0 by itself gave Z2 = 112. Adding together the 0 + and 4 + members of the ~74yb ground state band did not give as great an improvement (X z = 30) as the 0 + and 2 + combination. To test whether this improvement was fortuitous, the 16syb ground state (the L = 0 most similar to the 122 keV state) was also fit with the same combination of states, TABLE 5a States in 171yb This experiment Ex (keV)
0 75 94 123 168 208 228 248 318 369 448 486 502 601 778 831 907 940 960 995 1027 1048 1085 1111 1140 1198
Other experiments
~tr . . . . ") (#b/sr)
L
5 6.8 0.7 5 ¢) 27.7 1.6 7 9.4 1.0 5 484.6 5.5 5 6.9 0.6 5 90.2 2.1 5 c) 6.7 1.1 5d) 18.6 1.6 5~) 58.1 2.1 7 1.7 0.3 5 13.0 1.0 5a) 2.8 0.8 7 2.3 1.1 5 a) 7.0 0.6 7d) 1.7 0.5 7 2.2 0.4 5 a,~) 14.7 1.6 5 6,0 0.7 5 29.3 1.7 7 17.9 1.2 5 17.0 1.1 5 7.1 0.8 7 14.4 0.8 7 10.6 0.8 5 ~,d) 6.4 0.7 7 3.9 0.5
2
0 2 (4) 2 (4) (4)
(4)
E,, J'rrK" Ref. (keV) b)
0 1--1 75 5--1 95 7+7 122 5--5 168 9+7 208 7--5 230 7--1 247 9--1 317 9--5 369 13+7 450 11--5 487 1 1 - 1 509 13-1 604 13-5 779 15-5 832 15-1 902 3-3 945 1-1 995
3-1
•052 5-1 1079 7-3 1114 13-11 1144 7-1 1189 19-5
57) 27) s7) 27) ST) sT) sT) SS) sT) 2s) 2s) 2S) SS) 2S) S8) 5S) 6O) 60)
Other experiments
This experiment
Ex (keV)
1 2 4 2 7a) 1265 5 a) 1338 7d) 1389 5 1436 7 1487 7 1513 5e) 1 5 9 0 5e) 1624 5 1649 7 1673 7d) 1698 7 1736 7 1763 7 1827 7 1845 7 1868 7 1902 10 1966 10 60) 1995 10a) 2108 10 60) 2286 10 60) 2303 10 SS) 2373 10 60) 2476 10 2S) 2642 10
~,Ge.m" a) ~b/sr) 6.3 2.9 2.7 46.9 8.0 7.3 65.2 17.7 9.9 8.2 1.9 9.1 5.0 7.3 5.8 6.2 4.2 4.1 6.7 18.6 13.8 8.0 5.2 11.4 13.9 6.7
0.7 0.6 0.6 1.9 0.8 0.9 1.9 1.0 0.9 1.0 1.3 1.0 0.9 0.8 1.0 0.9 1.3 1.6 1.0 1.2 1.3 1.4 1.0 1.2 1.1 1.4
L
Ex J'rrK" Ref. (keV) b)
1266 15-11
ss)
1436 17-11
ss)
1626 19-11
5s)
1834
5s)
(2)
0
(0)
21-11
248
M . A . OOTHOUDT AND N. M. HINTZ TABLE 5b States in 169yb This experiment Ex
N a . . . . a)
(keV)
(/zb/sr)
27 5 71 5 87 5 99 5 194 7 245 5 268 5d) 484 10 516 5 570 5 659 5 750 5 808 5 1033 5 1108 5 1169 5 1350 5 e)
349.0 6.4 43.1 130.0 1.6 11.5 27.6 1.7 3.5 10.4 33.4 5.5 17.3 9.7 6.1 4.9 27.8
4.6 1.5 5.7 8.4 0.3 1.6 2.9 0.2 0.4 0.8 1.0 0.6 0.7 0.5 0.4 0.5 1.1
Other experiments L
0 2 2 (4) 4
2
Ref.
Ex
J'~rK"
(keV)
b)
24 71 87 99 191 244 264 487 512 570 660 747 807
1-1 9+7 3-1 5-1 5-5 7-1 9-1 11-1 13-1 5-5 3-3 9-5 15-1
59) 59) sg) 59) sg) 59) s9) s9) 59) 61) 61) 61) sg)
1111
5-5
61)
1353
3-1
6z)
This experiment Ex
Z a . . . . a)
(keY)
(#b/sr)
1426 5 1463 10 1479 5 1513 10 1528 5 1582 5 1621 5 1714 5 1740 5 d'¢) 1785 5~) 1943 7 1971 7 1997 7 d) 2053 12 2129 12 2286 12
10.8 2.0 4.3 5.3 9.8 20.2 9.3 7.4 14.0 13.5 10.7 9.7 4.7 10.4 9.0 6.3
0.7 1.4 0.7 1.1 0.8 1.5 0.9 1.2 0.9 1.1 1.3 1.4 1.3 0.8 1.1 1.5
Other experiments L
E~
J'~rK" Ref.
(keY)
b)
1466
7-1
61)
1616
3-
64)
0
(0) (o)
") Sum of center of mass cross section over standard angles. b) j , = 23, K" = 2 K .
°) Possible doublet. d) Contaminant subtracted. e) Doublet.
giving X2 = 57 for the 174yb ground state by itself and 48 for the sum of the 0 + and 2 + states. Since the improvement in Zz is so much more dramatic for the 122 keV state than for the ~6Syb pure L = 0 state, we conclude that the 122 keV state does have a significant amount (15-20 ~ ) o f L = 2 in it. In general the strongest L = 0 transfer for odd-N nuclei goes to a one-quasiparticle state with the same quantum numbers as the target ground state. The (p, t) amplitude is then a coherent sum over zero-coupled pairs o f neutrons picked up from the even-N core. The strong L = 0 states are at 122 keV in 171yb and 24 keV in 1 6 9 y b . The strength of these states is respectively 54 ~o (including any L = 2 and 4) and 37 (pure L = 0) of the average strength of the ground state transitions in neighboring even-N nuclei. A similar loss of strength in the odd-N Sn isotopes has been attributed to blocking by the unpaired neutron s 5). The (p, t) reaction may be expected to populate one- and three-quasiparticle states and states formed by coupling a quasiparticle (the unpaired neutron in the target ground state) to vibrations of the even-N core. Solov'ev et al. 56) have calculated the mixing of one-quasiparticle states with quadrupole and octupole vibrations of the even-N core. We will use their descriptions for the states throughout this section. The notation for these bands is f2[NnA] ® Q(2, #) where the first factor is the
(p, t) REACTIONS
249
asymptotic quantum numbers of the single-particle state and Q(2, #) is the vibration with multipolarity 2 and projection on the symmetry axis #. The bands thus formed have K = O__/~. The data are presented in table 5 and figs. 20 and 21. In the table excitation energies and errors (keV) are given in the first two columns. Columns 3 and 4 give the sum of center of mass cross section over standard angles and its relative error. Column 5 gives dominant L-values assigned in this experiment using CURVEFIT. Columns 6 and 7 give excitation energies (keV) and assignments from other experiments. The quantum numbers for the assignments in column 7 are J ' = 2J and K' = 2K. "Contaminant subtracted" means a state from another Yb isotope has been subtracted to give the cross section indicated. In the following paragraphs we discuss the bands observed. 4.3.1. The ½1521] band. In 171yb the bandhead (which is also the ground state) has a summed cross section of 7 #b/sr even though the I - A selection rule inhibits population of the state (subsect. 4.1). A possible mechanism for exciting this state is to pick up a neutron pair coupled to L = 2, leaving the even-N core in a v-vibrational state coupled to the unpaired neutron: 31512] ® Q(2, 2). Calculations s6) indicate that such a configuration contributes less than 1 ~ to the band. Members of the band up to perhaps the ~5 state were identified by comparison with known excitation energies 57, 5s). The weak _1~ member of the band would not be resolved from the ~5_member of the ~ [633] band. In 169yb the lowest strong L = 0 transition goes to the bandhead of this band at 24 keV. Band members up to perhaps ~5_ were identified by comparison with (e, 2n) data 59). The peak identified as the -~ member may contain strength from the member of the ~ [512] band, while the .~5_peak may contain significant strength from the ~ member of the 31521] band. 4.3.2. The -~[633] band. The ~, 9 and ~ - members of this band are seen in 17zyb. The ~ - and ~ - members would not be resolved from other states. In 169yb this band is the ground state band. The bandhead is not observed in agreement with the I - A selection rule. The ~ member of the band at 71 keV is populated but is very weak and is not well resolved from the much stronger states at 87 and 99 keV. A state near the expected 59) position of the _~3_member of the band at 270 keV is probably the 9 member of the ½1521] band since it is unlikely that the .½a_member of the band (which is populated by L = 7) would be observed when the @ member (L = 5) is not observed. 4.3.3. The 31512] band. This is the ground state band in 173yb and therefore gives the strong L = 0 transfer in the (p, t) reaction to 171yb as discussed above. Band members are seen up to the ~ state. The ~ member of the band appears to be a doublet in the (p, t) spectrum, although there are no other known states in this energy range. In ~69yb the bandhead is observed even though its population is inhibited by the I - A selection rules. The ~ band member would not be resolved from the 29-member
250
M. A. OOTHOUDT AND N. M. HINTZ
of the ½1521] band. A peak is observed near the location of the ~ - member of the band but since lower members of the band are not seen, it is probably due to the i a member of the ½1521] band. 4.3.4. The ~[521] band. Peaks at 907 and 1085 keV in 171yb are near the previously reported 6 o) ½ and ~ members of the a [521] band. The identification of these peaks is in doubt, however, since the s and ~ members of the ~ [523 ] are near these energies. Furthermore, the ~ member of the -~[521] band (which should appear near 981 keV assuming an I(I+ 1) spacing for the band) does not appear. In 169yb candidates for the s and ~ members of the band are observed at 659 and 808 keV. Calculations 56) show this band has a 4 ~o ½1521] ® Q(2, 2) component which might be expected to be populated in (p, t) through excitation of the v-vibration of the core. The s state 61) at 721 keV is not observed. The peak identified as the state may contain some strength from the ~ member of the 5 [523 ] band. The ~ member of the ~ [521 ] band may contain strength from the _1~_member of the ½[521] band. The bandhead of the lower ½1510] band is also near the ~ member of the ~[521] band 62), but the transition to the ½ state is 27-A hindered and no higher members of the ½1510] band are seen. Thus the ½ state would not be expected to contribute Strongly to the peak. 4.3.5. The ½1510] band. In a71yb population of this band requires pickup of a particle from a level well above the Fermi surface. Occupation of such levels may be expected to be low, so cross sections might be expected to be small. However, calculations 56) show that this band is 51 }o -~[512] ® Q(2, 2). This component might be populated strongly by picking up a pair coupled to L = 2 from the target, leaving the core in a v-vibration that is coupled to the unpaired s[512] neutron. Members of the band up to ~ are observed; the ~ member may contain strength from the -t2-smember of the ~-[505] band. The sum of band members seen is 37/~b/sr, which is an upper limit for the total L = 2 strength for the band. The strength of the v-bandhead in the neighbouring even-N nuclei is about double this. This loss of strength is expected from the splitting of the strength between the ½ and -~ states formed by coupling the vibration to the -~-. Further redistribution of strength may occur due to mixing of the resultant ½- state with the ½1510]. A candidate at 1389 keV for the second ½- state resulting from this mixing is discussed in subsect. 4.3.8. Members of the ~- band were not identified. In a69yb bandheads for fragments of the ½1510] state have been reported 62) at 813 and 1317 keV. At 808 keV in the (p, t) spectrum, a peak is seen with summed cross section of 17 #b/sr. Since the 2J-A selection rule inhibits population of the ½state and no higher band members are seen, this peak is probably the ~ member of the 31521 ] band and/or the -~- member of the ½1521 ] band. Peaks are observed at 1350 and 1463 keV near the energies of the -~ and ~ members of the 1317 keV band. The ½ member of the ½1510] would be inhibited by the Z-A selection rules, but the unobserved s member would be allowed. 4.3.6. The~}[523]band. In ~71Yb the 2~-A selection rule inhibits population of
(p, t) R E A C T I O N S
25i
the bandhead. A 15 #b/sr peak is seen near the expected energy of the bandhead (907 keV) but it may be the bandhead of the 31521] band. Peaks at 995, 1085 and 1I98 keV are reasonably dose to the previously reported 6a) ~, 29_and - ~ members of the 3[523] band at ~ 987, 1083 and 1208 keV, although other states couldcontribute to these peaks. The 3 and 9 members of the ~ [523 ] band are observed in ~69yb at 570 and 750 keV. The ½ member has been previously reported 64) at 648 keV; the nearby bandhead ofihe 31521] band may obscure this state. 4.3.7. The -111505] band. Data from the (a, 3n) reaction s8) indicate the presence 11 15 of an J~-[505] band in 171yb with members at 981 (-~-), 1114 (~-), 1266 (-~), 1436 (-~-), 1626 (~-) and 1834 (21_)keV. In the (p, t) spectrum peaks are seen very close to all these energies except for the ~-~ bandhead. This band has not been identified in 169yb" 4.3.8. States based on collective vibrations o f the even-N core in 171yb" An L -- 0 transfer a t 1513 keV ~ndicates a ~- state with 13 ~ of the summed strength of the 3- at 1:22 keV. To find higher band members we use an inertial parameter of hZ/2J 11.4 keV similar to that of the ~[512] band. We find candidates for a at 1590, a 9 at 1698, an ~1 at 1827 and a _13_at 1966 keV. It is possible, however, that the 1698 and 1827 keV peaks belong to other bands. Since the ~- Nilsson orbitals near the Fermi surface have already been accounted for, this band probably is a K = 0 Vibration of the core coupled to the ~[512] unpaired neutron. Surprisingly, the 1513 keV state is considerably stronger than the excited K = 0 bandheads in adjacent even-N nuclei. A strong, possible L = 2 transfer to a peak at 1389 keV indicates a state with total angular momentum of k, 3 , . . . , or 9. A low value for the angular momentum of the state is favored by the fact that the ~72yb(d, t) reaction 60) populates the state whereas the t72yb(3He, a) reaction 63) does not. It is interesting to speculate that this state may be the other K = ½ band resulting from mixing the ½1510] with the ~[512] ® Q(2, 2), as discussed in subsect. 4.3.5. The 1436 and 1487 keV states could be the 3 and 3 members of the band although they give inertial and decoupling parameters somewhat higher than average for vibrational states in this region 66). Peaks seen at'1590, 1673 (or 1698) and 1868 keV could be higher band members. 4.3.9. Other states in 169yb. Three weak transitions that may be L = 0 are observed at 1513, 1582 and 1740 keV. Sincethe target is ½-, these states are necessarily ½-. Assuming inertial and decoupling parameters similar to other nearby ½bands, states at 1621 and 1785 keV could be the 3 members of the 1582 and 1740 keV bands respectively. 4.4. N A T U R A L T A R G E T S
Even-Z nuclei from Gd through Os were investigated to find 0+ states strongly populated by the (p, t) reaction. Targets consisting of elements with the natural abundances of isotopes were exposed to the i9 MeV proton beam and tritons were
252
M . A . OOTFIOUDT AND N. M. HINTZ
collected with nuclear emulsion plates. D a t a were t a k e n only at 27½ ° near the L = 0 m a x i m u m . The lowest L = 0 transfer for each nucleus was identified using twon e u t r o n s e p a r a t i o n energies ( $ 2 , ) f r o m mass m e a s u r e m e n t s 29, 67-71) a n d intensity systematics. Excited states were identified using k n o w n level schemes. F o r e v e n - N nuclei only m e m b e r s o f the g r o u n d state b a n d u p to 4 + o r 6 +, 7-vibration b a n d h e a d s , 3 - o c t u p o l e states a n d excited 0 + states were identified with a n y certainty due to the
(p,t) GD
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NATURAL ER
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RESIDUAL NEUTRON NUMBER Fig. 22. Cross sections for states populated by the (p, t) reaction at 27½° and Ep = 19 MeV using natural targets (except for Yb and Hf). Cross sections are upper limits only. The upper figure shows ground bands, while the lower shows excited bands. For excited 0 + states, the sum of all known 0 + states observed is plotted. Relative errors are shown only where they are larger than the data points. Points plotted are 0(0+), A(2+), 11(4+ or 3-) and ×(6+). h i g h density o f states p o p u l a t e d . F o r o d d - N nuclei, only the lowest L = 0 transfer was identified; it was assumed t h a t the lowest L = 0 transfer goes to the one-quasiparticle state with the same q u a n t u m n u m b e r s as the target g r o u n d state. The quasiparticle assignments o f B u n k e r a n d Reich 66) are used here unless otherwise noted. Cross sections for the states o b s e r v e d are p l o t t e d in fig. 22. F o r completeness, d a t a t a k e n with isotopic Y b a n d H f targets are also shown, In general the cross sections are u p p e r limits only, since the high density o f states could p r o d u c e p e a k s
(p, t) REACTIONS
253
composed of states from different nuclei. Relative cross sections between isotopes of the same element were obtained using the natural abundances of the isotopes 17). Normalization between elements (except for Os) was obtained using cross sections from recent (p, t) experiments 4, 5, 7) at 17.5 and 18 MeV. No corrections were made for the differences in proton energy, since DWBA calculations similar to those in sect. 5 predict a change in the cross section for 0 + states near the L = 0 maximum of less than _+ 10 ~ over a range of proton energies from 17 to 20 MeV. For Os, elastic proton scattering was done at 50 °, 60 ° and 70 ° and the cross sections were assumed to have the same ratio to Rutherford as 176yb(p, p). It is not expected that the accuracy of the normalization between elements is better than 25 ~ . The cross sections obtained for the lowest L = 0 transfers are given in table 2 along with Qvalues measured. Since the thickness of the targets and carbon backings were not carefully measured, absolute Q-values could not be obtained, although relative Q-values should in general be good to _ 5 keV. The measured Q-values for all isotopes of an element were shifted to agree with accurately known Q-values 29, 68, 69). The relative cross sections measured agree reasonably well with recent data taken with isotopic targets 4- 7, 72). The previously uninvestigated Os isotopes do not seem to have any excited 0 + states with greater than 10 ~ of the ground state strength up to an excitation energy of 2.5 MeV. In the paragraphs below, contaminants in the targets are discussed before the rare earth elements of interest. Cd The natural Os target contained a considerable amount of natural Cd. The presence of the Cd peaks allowed an independent check of the Q-value determinations. The Q-values were shifted for this target to give best agreement with Szn values 6s) for 192Os and 19°Os. The resulting San values for the Cd isotopes (table 2) are in good agreement with previously determined values 71). Ta The Gd, Dy and Er targets contained a significant amount of Ta from either the Ta powder used to reduce the oxides or from the Ta evaporation boat. A plate was exposed with a pure Ta target at 27½° to identify possible contaminant states. The major Ta states seen up to 2.5 MeV and their strengths relative to the ground state are 0 keY (1.00), 137 keV (0.06), 296 keV (0.03), 1520 keV (0.06), 1549 keV (0.01), 1732 keV (0.02), 1892 keV (0.01), 1975 keV (0.02), 2134 keV (0.01) and 2337 keV (0.02). The 0, 137 and 296 keV states have been previously identified 73) as members of the K ~ = 7+ ground state band. The 6 ~ 1520 keV state might be a K = 0 core vibration similar to the 4 ~ state excited by (p, t) at 1776 keV in the even-Z isotone 178Hf. By assuming that ratio to Rutherford for Ta is the same as for 176yb(p ' p) and doing elastic scattering at 27½°, an absolute cross section of 354 #b/sr (_+25 ~ ) was obtained. This is somewhat lower than the adjacent even-Z nuclei. If this value is correct, it indicates the odd proton has a surprisingly large effect on the neutron pairing. Such a reduction has been seen 23) for 1°9Ag(p, t) relative to 10Spd (p, t). Au One of the W targets was made on an Au backing. In a plate exposure with
254
M.A. OOTItOUDT AND N. M. HINTZ
a pure Au target, the major states observed below 2 MeV and their relative strengths are 0 keV (1.00), 261 keV (0.02), 550 keV (0.03), 828 keV (0.02), 1349 keV (0.01) and 1599 keV (0.02). The 0, 261 and 550 keV states have been previously identified 74) as -}+, ~+, and ~+ or 5+ states. The 1599 keV state might possibly be a K = 0 core vibration similar to the 4 ~o state at 1551 keV observed 7Z) in the (p, t) reaction to the isotone a94pt. An absolute cross section of 454 #b/sr (_+25 ~ ) was determined by the procedure used for the Ta target. G d - The absolute cross sections for the Gd isotopes were obtained by comparison with the 16°Gd(p, t) ground state cross section measured by Fleming e t al. 4) at 18 MeV. The Q-values were adjusted to give best agreement with S2n values 69) for 158Gd and 16°Gd. The lowest L = 0 transfer in 15aGd to the 31521] ground state 75) was not resolved from the strong 0 + state at 1050 keV in 156Gd. The 0 + state was subtracted from the peak to give the 153Gd cross section using the strength relative to the ground state given in ref. 4). No strong excited states other than those identified in refs. 4, 6) were observed. D y - Absolute cross sections were obtained by comparison with the cross section given by Maher e t al. 5) for the ground state transition in 164Dy(p, t) at 17.5 MeV. The Q-values were shifted to give agreement with S2n values 69) for 164Dy. The S2n value for 16~Dy disagrees strongly (18 keV) with the previous value 69). Since there are no other peaks in the proper energy range with sufficient strength for the L = 0 transfer, either the previous S2n value is incorrect or the excitation energy (178 keV) of the -}[642] state 66) is wrong. The ground state of the Ta contaminant obscured the 4+ member of the 16ZDy ground state band. Three relatively strong peaks were observed that do not correspond to states in the even-N isotopes observed in ref. 5). They could be states in 16~Dy at 1090, 1128 and 1270 keV with approximately 20 of the lowest L = 0 strength or states in ~S9Dy at 529, 567 and 709 keV with approximately 25 ~o of the lowest L = 0 strength. However, these peaks could equally well be sums of unresolved weak states. E r - The cross sections were normalized using the ground state transition cross section for 166Er(p, t) measured by Maher e t al. 5) at 17.5 MeV. The Q-values were shifted to give best agreement with the ~68Er ground state and the ~66Er first excited state San values 69). The 166Er ground state was obscured by the Ta ground state. A cross section for the ~66Er ground state was obtained from the measured ratio of the ground and 137 keV states of Ta. The a62Er ground state was obscured by the 1520 keV Ta state. The Q-value for 164Er(p ' t) was obtained from the first excited state and the cross section obtained by using the known cross section ratio of the 137 and 1520 keV states. The'lowest strong L = 0 transfer for 167Er(p, t) should go to the previously unobserved ~ [633 ] state. However, due to strong Coriolis mixing of the positive parity bands from the i~ spherical orbit, the 5 + state at 63 keY, which is mainly a rotation, based on the ~[642] band 76), may be expected to have appreciable (p, t) strength. In the (p, t) spectrum, a peak with a 96 #b/sr cross section appears near the expected location of the 63 keV state. A strong (293 #b/sr) peak at
(p, t) REACTIONS
255
462 keV is not identified with any known Er states and may therefore contain a major part of the ~[633] bandhead. In fig. 22 the sum of these two states is plotted. Preliminary (p, t) data 77) using an isotopic 167Er target Confirm that the 63 and 462 keV peaks are both states in ' 65Er populated by L = 0 transfers. Three strong peaks have not been identified with known Er states. They could be either states in 16SEt at 3016 keV (with 11 ~o of the ground state strength), 3057 keV (6 ~ ) and 3100 keV (20 ~ ) or states in 165Er at 1373 keV (with 18 ~ of the strength of the 462 keV state), 1414 keV (10 ~ ) and 1457 keV (32 ~). H f - Absolute cross sections were obtained from the 178Hf(p, t) data of this experiment. The Q-values were shifted to give best agreement with S2n values 29, 67) for the ground state transitions for 1s OHf and 178Hf(p ' t) and the transition to the 321 keV state in 177Hf. The S2n values for the 177, 176Hf(p ' t) reactions have been previously determined to an accuracy of only 42 and 59 keV respectively z9). The lowest L = 0 transfers for 177, 176Hf(p ' t) were found to lie within 20 keV of each other, but on the basis of cross section systematics and the location of the first excited states in the respective bands, it is possible to arrive at the Q-value assignments given in table 2. The (p, t) S2n value for 177Hf disagrees strongly (132 keV) with the previous value 29). A strong peak not identified with any known H f states could be a state at 1478 keV in a77Hf with 21 ~ of the lowest L = 0 strength or a state at 656 keV in 175Hf with 10 ~ of the lowest L = 0 strength. W - Absolute cross sections were obtained by comparison with the ground state transition cross section at 18 MeV for 186W(p, t) given by King et al. 7). A cross section determination similar to that done for the Ta target gave a normalization about two-thirds of the King value, in better agreement with the systematics for ground state cross sections (fig. 22). However, in view of the uncertainties in the proton-scattering method, we will use King's value here. The Q-values were shifted to give best agreement with S2n values 68) for 184W. The S2n values for 183W and 82W are not well known (see table 2) and identification of the L = 0 states for these nuclei was difficult. The final identification was made on the basis of other weak excited states in the residual nuclei. The values of $2. which we obtain do not agree well with previous values. The peak chosen as the L = 0 for 183W(p, t) is surprisingly weak, which may indicate an incorrect identification of the state or a real loss of strength due to blocking of the ½[510] orbital with its large (p, t) strength (see sect. 5). If the assignment is incorrect, the only possible alternative choice giving a reasonable strength to the aSaW(p, t) L = 0 transfer is to make it and the ~s2W(p, t ) l S ° w ground state transition an unresolved doublet. This choice, however, makes the 18 oW ground state quite weak. O s - Cross sections were normalized as desciibed above. The Q-values were shifted to give best agreement with $2. values 68) for 1920s and 19°Os. The value of $2, obtained here for 188Os does not agree well with previous values. The i87Os and 186Os(p, t) reactions were not identified. The peak identified as the v-vibration in ~86Os at 766 keV is unusually strong (19 ~ of the ground state strength) compared
256
M . A . O O T H O U D T A N D N. M. H I N T Z
to other y-vibrations. It is possible that this peak could contain strength from a state in another nucleus, but this would give as upper limits a 4 % state at 1734 keV in 19°Os, a 10 % state at 1334 keV in 188Os or a 25 % state in 187Os at 1138 keV. It is possible that strong Os states are obscured by the Cd contaminant peaks.
iO s
L
"TeYb(p,t) Ep =lgMeV CC
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102 ==~', ,,,,.
-~
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60
90
120
150
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The W and Os nuclei span the shape transition from deformed to spherical near N = 110. For the Sm nuclei the shape transition near N = 88 results in 0 + states excited by (p, t) with over 57 % of the ground state strength 1). In the W and Os nuclei, however, the strongest 0 + states observed have less than 10 % of the ground state strengths. The lack of strong excited 0 + states may be due 72) to the more gradual shape change in this mass region than near N = 88. 5. Calculations 5.1. G R O U N D BAND INTENSITY RATIOS
Because of strong inelastic excitation of rotational states, it is necessary to use a coupled-channel treatment of transfer reactions in deformed nuclei. Tamura et el. 9) have used coupled-channel equations to calculate cross sections for 176yb(p, t) to
(p, t) REACTIONS
257
members of the ground band of 174yb" Ascuitto and co-workers 1o, 12, 13) have used the "source-term" method 84) for (p, t) reactions. This method does not use the exact CCBA equations but it is believed to give the same results 9). The calculations of Tamura et al. 9) agree well with those of Ascuitto et al. lo, 12), so we will discuss only the latter. Ascuitto, Glendenning and Sorensen 1 o, ~2) adopted the Bohr-Mottelson adiabatic hypothesis to write wave functions. The intrinsic wave functions were BCS wave functions constructed from the single-particle eigenfunctions of a deformed WoodsSaxon potential with a shape defined by the deformation parameters f12, f14 and fit. The deformation constants for the protons, tritons and bound neutrons were separately scaled from values obtained from analysis of c~-inelastic scattering 1 s) to ensure the same multipole fields for all particles. The same values of flz were used for both 176yb and 174yb. The BCS calculations were done with the 20 levels nearest the Fermi surface. The strength of the pairing force, G, in the BCS calculations was adjusted to reproduce a gap parameter o f A , = 0.36 MeV for 176yb and 0.575 MeV for 174yb instead of the experimentally 29) observed odd-even mass differences, Pn = 0.593 and 0.687 MeV respectively. Ascuitto et aL 12) suggest that having An < Pn may be necessary due to the low density of single-particle levels in this region and to obtain a smooth transition to the more neutron deficient Yb isotopes. Optical parameters were obtained from fitting inelastic proton data for 176yb from this experiment (sect. 3) and the 20 MeV elastic triton data of Flynn et al. 85) extrapolated to Yb values. The only remaining parameter is a normalization constant which is the same for all states. Fig. 23 shows the fits obtained 12) to preliminary data from this experiment. The CCBA calculations fit the data extremely well, especially when compared to concurrent DWBA calculations which give reasonable agreement with 0 + and 4 + cross sections but badly overestimate 2 + cross sections and underestimate 6 + cross sec, tions. Furthermore the DWBA calculations do not reproduce the 2 + shape well. The critical feature of the calculations is the multi-step processes not included in DWBA but which are explicitly part of the CCBA calculations, For the 2 + state the main two-step processes are (i) inelastic excitation of the target by the incident proton to the ground state band 2 + followed by pickup to the 2 + in the residual nucleus, or (ii) pickup to the residual ground state followed by inelastic excitation to the 2 + by the exiting triton. Ascuitto et aL i o, ~2) show that the strengths of these two-step processes are only slightly less than that of the direct route, that all three routes have angular distributions with shapes different from each other and the data, and that all three routes overestimate the magnitude of the cross section. However, the three processes interfere destructively to give a good fit to the data. 5.2. EXCITED K = 0 BAND INTENSITIES Abdulvagabova and Pyatov l l ) have done PWBA-RPA calculations with ordinary pairing forces to calculate K = 0 bands in the Yb isotopes. They predicted excited
258
M. A. O O T H O U D T
A N D N. M. H I N T Z
0 + states in 174yb at 1.45, 2.00 and 2.58 MeV with 16, 11, and 5 % of the ground state strength in the (p, t) reaction. These states correspond well in strength with the experimentally observed states at 1.489, 1.886 and 2.100 MeV with 26, 12 and 7
,
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Fig. 24. C o u p l e d channel (CC) calculations for t h e K = 0 b a n d calculated at 1.47 M e V a n d experimentally observed at 1.489 MeV. T h e n o r m a l i z a t i o n h a s been determined by fitting the g r o u n d state. Results o f D W B A calculations are also s h o w n . T h e theoretical curves were t a k e n f r o m fig. 8 o f r e f . 13). T h e d a t a points are f r o m this work.
0
60
120
180
OC.M.
Fig. 25. C o u p l e d c h a n n e l calculations for the K = 0 bal~d in ~74Yb calculated at 0.93 M e V a n d experimentally observed at 1.886 MeV. NormalizatiorL is fixed by fitting t h e g r o u n d state. T h e theoretical curves were t a k e n f r o m fig. 9 o f ref. la). T h e d a t a points are f r o m this work.
of the ground state strength. However, for 172yb ' 170yb and 168yb ' states between 1.5 and 3.5 MeV were predicted with 20 to 35 ~ of the ground state strength. Experimentally no such strength is observed UP to 4 MeV of excitation energy. Abdulvagabova and Pyatov 11) suggested that quadrupole forces may be necessary to fit experimental results. Further calculations have been carried out by Ascuitto and Sorensen 13) for excited K -= 0 bands. A more rapid method was used to calculate the form factors
(p, t) REACTIONS
259
than previously 1 o, lz), but in principle these calculations differ from those discussed in subsect. 5.1 only by the addition of an ordinary surface-peaked quadrupole force. The excited K = 0 bands have only a few quasiparticle components, so their energies and strengths are strongly dependent on the energies of the single-particle levels. Due to the lack of accurate information on single-particle energies, Ascuitto and Sorensen ! 3) chose to use levels generated in a Woods-Saxon well using parameters from the lead region. Thus they did not expect to fit either the experimental energies or strengths for individual states, but they did expect to obtain the correct total 0 + strength. The strength of the pairing force was adjusted as in the ground band calculations lo, lz). In order to fix the strength of the quadrupole interaction, the average energies of low lying K = 0 bandheads and the B(E2) and p(E0) values from these bands to the ground band or within these bands were fit. The agreement with the electromagnetic data is good. In the reaction calculations for excited K = 0 bands, the 0 +, 2 + and 4 + members of the target and final K = 0 bands were included in the coupling. Inelastic excitation of the excited K = 0 bands from the ground state was not included, since interband B(E2) values are so small that the inelastic processes would result in a two-step cross section two orders of magnitude weaker than the direct route 13). The calculations reproduce the ratio of the sum of the excited 0 + strength to the ground state for a76yb(p, t) quite well: 46 ~ predicted compared to 45 ~ experimentally. Fits to preliminary data for the 1489 and 1886 keV K = 0 bands in 174yb are shown in figs. 24 and 25; data from ref. 8) have been plotted on these figures. G o o d results were also obtained by Ascuitto and Sorensen 13) for 16°Gd, 174yb and 186W(p, t). The sharp decrease in strength for 174yb(p, t) to excited 0 + states was reproduced well in the calculation. In spite of the success of the Ascuitto-Sorensen calculation in fitting the general trends of excited 0 + strength and angular distributions in ground and excited K = 0 bands, several questions remain. The BCS calculations of Ogle et aL 46), for instance, do not converge near 176yb ' i.e., the level density is so low that the BCS equations have no solution for a reasonable value of G. Thus it is not clear that the wave functions used in refs. 1o, 12, 13) are correct. In addition, it is not clear whether the strong 0 + transitions are caused by gaps in the single-particle spectrum, as discussed in subsect. 4.2.2, or by the transfer properties of individual Nilsson levels. For example, if the transfer amplitudes from different orbitals are unequal, excited 0 + strength can occur even without gaps in the single-particle spectrum s6). To see this, consider a simple mode1 with two active orbitals at the Fermi surface. In the 176yb ground state, denoted I176) g, both levels are occupied. In 174yb, 0 + states can be written as linear combinations of pairs of holes in orbitals 1 and 2: 1174) g = [aA1 + bA2] [176) g, ]174)* = [bA 1 - aA2] [176) g,
260
M . A . O O T H O U D T A N D N. M. I t I N T Z
1 and Ai destroys a zero coupled pair in orbital i. If ai is the L = 0 (p, t) amplitude for orbital i, then the cross section ratio for the two 174yb states is
where
a 2 q-b 2 =
R
--
~*
lalb-%al 2
6 g
[aaa+(12b[ z
When the transfer amplitudes are nearly equal, strong configuration mixing (a ~ b) destroys the excited 0 + strength. If, however, al >> a2 (or the converse) large excited s t r e n g t h can be observed even when a ~ b. For example, if a z = 5a 1 and a ~ b, R = ~. For the pure configuration case (a = 1, b = 0), R = 25. Thus without using gaps in the single-particle spectrum or invoking any special collectivity, we have an excited 0 + state populated by (p, t) with between 0.44 and 25 times the ground state strength. It is desirable to make a more quantitative exploration of the effects of gaps, level ordering and transfer amplitudes on the excited L = 0 strength. We have used a pairing force approximation to calculate 0 + states and the DWBA to calculate (p, t) cross sections for states in 174yb. We did not perform CCBA calculations, since refs. lo, 12, 13) show that DWBA is nearly as good as CCBA for reproducing the shapes and relative intensities of 0 + states. An exact diagonalization of the pairing force was performed with the computer code S P R I N G 87, 88) to avoid problems with BCS non-convergence. The G-values and energies of twelve Nilsson levels for 175yb from ~ [642] through J~-[615] (see fig. 16) were taken from ref. 46). The proper number of pairs for the nucleus considered were placed in these levels in all possible combinations. For most calculations the pairing force was diagonalized only among the 30 states with the lowest summed Nilsson single-particle energies. These 30 states span an energy range of 5 to 6 MeV. The eigenvalues and eigenvectors then give excitation energies for 0 + states and occupation probabilities (V 2) for the Nilsson levels. Spectroscopic amplitudes (X) for (p, t) and (t, p) were then computed from the eigenvectors for the target and residual nucleus. The X were converted from a Nilsson basis to spectroscopic amplitudes in a spherical harmonic basis (S ~) using the Nilsson coefficients 89) for a deformation parameter of t/ = 0.6. The S ~ were used in T W O P A R 87, 9o), a zero-range two-nucleon transfer DWBA code. In the DWBA calculation the single-particle bound-state wave functions were calculated in a Woods-Saxon well with a diffuseness of 0.65 fm, a spin-orbit coupling of 25 times the Thomas term and a radius of 1.25 A ~ fro, where A is one mass unit less than the target. The depth of the well was adjusted for each single-particle wave function to give a binding energy equal to half the two-neutron separation energy in table 2. Note that the S ~ are calculated for a spherical harmonic oscillator well, whereas a spherical Woods-Saxon well is actually used for the reaction calculation. The optical model parameters for the protons were set 5 in table 3. For tritons, the parameters of Flynn et al. 85) for elastic scattering on 182W were modified according to the mass and energy dependence of the Beechetti-Greenlees global triton param-
261
(19, t) REACTIONS eters 91) to Y b values:
vR=
183.1-6.4 (¢-~o)-0.17(E-20)
MeV,
w~=
21.6-110.(4-~o)-0.33(E-20)
MeV,
wo=
0,
FR
1.16 A ~ fro,
a~
q = 1.332 A ~- fro,
= 0.76 fm,
a I = 0.994 fm,
where ~ = (N-Z)/A, 4o is the value o f ~ for 182W and E is the energy o f the outgoing triton for the g r o u n d state o f the reaction considered. TABLE 6
Magnitudes of spherical components a) and relative (p, t) strengths of Nilsson levels li
~[642] ~[523] ~[521]
8
~[6331
9
lh
2g
450
2 499 161
2f
86
35 65 1
54 374
170
26
148
163
~[512] ~[514] ~[624] ½1510] ~[503] ~[512] ~15]
89 586
36 18
8
556 30
55 34 94
9 12 7
238 523
136 654 84
4
530
~n~jb)
1
10
5
22
132
298
515
~rrel.
104
44
½15211
7
3p
3 60 0
488
3d
352
24
14
") Multiplied by 1000. b) Relative cross section at
01a b ~
73
221
13
404 80
469 1000
301
31 32 374 19 522
387 14 13 1000 500 370 11
714
27½°.
Using levels f r o m ref. 46), the energies o f the lowest four 0 + states are calculated to be 0, 1.974, 2.635 artd 2.694 M e V with relative cross sections at 27½ ° (near the L = 0 m a x i m u m ) o f 1.00, 2.33, 0.003 and 0.46 respectively. Experimentally, the excitation energies are 0, 1.489, 1.886 and 2.100 M e V with relative cross sections at 27½ ° o f 1.00, 0.25, 0.12 and 0.05. Thus the predicted 1.974 M e V state contains m o r e than five times as m u c h strength relative to the g r o u n d state as is experimentally seen in the sum o f all excited 0 + states. Since only relative cross sections are calculated, it is n o t clear whether the excited state cross section is too high or the g r o u n d state too low. Calculations using the lowest 56 states in the matrix diagonalization gave no significant improvement 92): excitation energies were shifted by less than 15 keV and the ratio o f the cross section o f the 1.974 M e V state to the g r o u n d state was reduced by less than 10 ~ .
262
M . A . OOTHOUDT AND N. M. HINTZ
To find where the excess strength comes from, the (p, t) strength from each of the Nilsson orbitals and Woods-Saxon single-particle levels was individually calculated. In table 6 the magnitudes of the spherical components t and the relative (p, t) cross section (last column) at 27½° are listed for each Nilsson orbital. Also listed at the bottom of the table (labelled a, zs) are the (p, t) cross sections for each of the spherical single-particle levels relative to the 3d~_. It is interesting that the 3d~ has a higher cross section than the 3p levels. In the lf~ shell the 2p~ level has the highest (p, t) cross section ss). Evidently for this mass region, the 3d~ wave function gives a better overlap than the 3p wave functions with the triton internal wave function in the form factor. Among the Nilsson orbitals, the ½1510] has by far the largest (p, t) cross section due to a large 3p~ component. The cross sections for different Nilsson orbitals vary by a factor of up to 100, easily giving the factor of 5 in amplitudes used in the example above. TABLE 7 Spectroscopic amplitudes for 0 + states populated by 176yb(p, t)lT*Yb E~
Nilsson orbital
(MeV)~[5211 415231 416421 ½15211 ½[633] 415121 -~[5141 3[624] ½15101 315121 -~[5031 ~[615] 0.000 1.974 2.634 2.693
0.052 0.034 0.070 0.042 0.022 0.011 0.082 0.041
0.034 0.041 0.011 0.040
0.086 0.206 0.120 0.776
0.083 0.185 0.092 0.499
0.117 0.936 --0.099 --0.277
0.934 --0.104 --0.048 --0.050
0.286 0.054 0.046 0.025
0.157 0.050 0.009 0.030
0.068 0.080 0.049 0.031 0.035 0.016 0.009 0.005 0.007 0.016 0.021 --0.012
The contributions of the various orbits to the spectroscopic amplitudes for the lowest four 0 + states are shown in table 7. The out of phase component of the 1.974 MeV level, 71514], has too weak a transfer amplitude to reduce the excited 0 + strength sufficiently. In order to reduce the ratio of the excited to ground 0 + cross sections in the example above, the mixing of the two levels must be increased. In the actual case presented here mixing may be increased by either increasing the value of G or by decreasing the gap at the Fermi surface**. Increasing G by a physically unrealistic factor of 2 from 0.132 MeV to 0.264 MeV improves the relative cross sections to only 1.00, 0.50, 0.001 and 0.04. This change is mainly due to an increase by a factor of approximately 4 in the ground state cross section due to an increase in the ½1510] occupation in the target. The excitation energies, however, have become worse: 0, 2.156, 2.865 and 2.984 MeV. The levels above the -~[512] (see fig. 16) were shifted downward together by as much as 400 keV to decrease the gap in the single-particle spectrum between the * The magnitude of a spherical component is defined as the absolute value of S~_ for an (nlj) component in the L = 0 pickup of a pair from a Nilsson orbital with X = I. ,t Cf. the discussion of t h e two-level pairing phase transition model of Broglia et al. 41, 44, 4s) in subsect. 4.2.2.
(p, t) REACTIONS
263
~[512] and the 71514] levels. The best agreement in cross section ratios is obtained for the 400 keV shift, which gives relative intensities of 1.00, 1.76, 0.009 and 0.44 and excitation energies of 0, 1.227, 1.878 and 1.941 MeV. Thus we have not been able to get good agreement with excitation energies and cross section ratios for large changes in either G or the gap in the single-particle spectrum. Furthermore, it may be expected that such large changes will cause disagreements with experimental oddeven mass differences and quasiparticle energies which were fit in ref. 46) to obtain energies for the Nilsson levels used here. It is interesting to note that the quasiparticle energy of the ½1510] orbital is known 46) to no better than 100 keV. Thus it may be possible to shift this level with its large (p, t) amplitude to give the ground state enough strength to dominate the excited states without destroying agreement with quasiparticle energies. Preliminary calculations 92) indicate that the ratio of the first excited 0 + state to the ground state and the excitation energy can be reproduced well by shifting the ½1510] down to slightly above the 71514] level. Further calculations are in progress and will be reported later. We conclude from these calculations that comparison with cross sections and energies for 0 + states excited in (p, t) may be a very sensitive test of proposed singleparticle level schemes. It must be emphasized, however, that the analysis presented here is quite crude. In particular, the effects of ordinary quadrupole and spinquadrupole forces may affect the strength of excited 0 + states 4o).
6. Summary and conclusions The (p, t) reaction has been used to study a number of nuclei in the rare earth region. The reaction has been shown to be good for identifying L = 0 transfers even with incomplete angular distributions. The reaction is not as reliable for determining other L-values. For even-N nuclei many states have been identified through comparison with known level schemes. Members of ground state rotational bands up to 8 + in some cases and 6 + in most have been observed with targets of is°I-If, 17SHf and all the stable even-N Yb isotopes. Calculations presented in the recent literature have shown that DWBA calculations do not fit the excited members of the ground bands nearly as well as do CCBA calculations. The existence of excited K = 0 rotational bands strongly populated by (p, t) has been shown. These bands are observed in 174yb and 176Hf' but are seen only weakly in other nearby nuclei. These bands have been discussed in terms of possible nuclear collective modes such as the pairing phase transition. CCBA calculations 13) with a surface-peaked quadrupole force have been discussed and shown to give a good account of the data. It has also been shown with DWBA calculations and simple pairing theory that a strongly populated excited 0 + state may result from Nilsson singleparticle orbitals with larger than average (p, t) amplitudes. It is not clear whether the strong 0 ÷ states are produced by collective modes, strength from particular
264
M.A. OOTHOUDT AND N. M. HINTZ
Nilsson orbitals, gaps in the single-particle spectra or some combination thereof. Further theoretical investigation of the 0 + states is necessary. A number of other bands have been observed in the even-N Yb and H f nuclei. Members of g a m m a and octupole vibrational bands have been populated with appreciable strength. In addition several previously/known two-quasiparticle states have been observed. It is interesting that several unnatural parity states, such as 3 + members of g a m m a bands, were observed with weak but nonzero cross sections. In view of the problems with calculations for 0 + states, no attempt has been made to do calculations for these states. The (p, t) reaction to states in the odd-N nuclei 171yb and 169yb has also been studied. The lowest L = 0 transfer has been found to proceed to a single-quasiparticle state with the same quantum numbers as the target ground state but with reduced intensity relative to the even-N nuclei. The L = 0 transfer to a state at 1513 keV in 171yb with 13 % of the strength of the lowest L = 0 transfer may indicate the presence of a K = 0 vibration of the even-N core coupled to the unpaired ~[512] neutron. Due to the high density of states excited and greater theoretical complications, considerably less information has been extracted for the odd-N nuclei than for even-N nuclei. Natural targets for elements in the rare earth region have been used to obtain information on trends in cross sections for members of ground bands, g a m m a bandheads, 3 - oetupole states and strongly excited 0 + states. In addition it proved to be rather easy to obtain $2, values that agree well with more ditfi~ult mass measurements. We have shown that the (p, t) reaction is a useful spectroscopic tool in the rare earth region. However, there is a need for more extensive experimental and theoretical studies. The use of Prof. Benjamin F. Bayman's computer codes T W O P A R and S P R I N G and several useful discussions are gratefully acknowledged. We also wish to thank Judy Gursky of the Los Alamos Scientific Laboratory for preparing the H f and Os targets, George Ott for preparing the other targets, Philip Debenham, Albert Kuhfeld, Ralph De!ong and Richard Wallen for aid in data taking and Marjorie Maloney, Mary Evenson and Christie Helgeson for scanning the plates. One of us (M.A.O.) wishes to express his thanks for support received during this work from a N A S A Traineeship (1968-9) and an N S F Graduate Fellowship (1969-72). References
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(P, t) REACTIONS
265
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THE
Nuclear Physics A213 (1973) 221 --266; (~) North-Holland Publishiny Co., Amsterdam N o t to be r e p r o d u c e d b y p h o t o p r i n t or microfilm without written permission f r o m the publi8her
(p, t) R E A C T I O N S
ON NUCLEI
IN THE
RARE
EARTH
REGION
MICHAEL A. O O T H O U D T t and N O R T O N M. H I N T Z J. H. Williams Laboratory, School of Physics, University of Minnesota, Minneapolis, Minnesota 55455tt Received 12 March 1973 (Revised 14 June 1973) Abstract: The (p, t) reactions on isotopic targets of 17s. 1s OHf and all the stable isotopes of Yb and on natural targets of Gd, Dy, Er, HI', Ta, W, Os and Au were studied at a beam energy of 19 MeV with an average resolution of 12 keV. A split-pole magnetic spectrometer was used to measure (13,t) Q-values and absolute differential cross sections. On the basis of angular distribution shapes definite 0 + and tentative 2 + assignments were made. Rotational bands were identified assuming an I ( I + 1 ) spacing. The (p, t) reaction populates excited 0 + states strongly in 174yb, 1 7 6 H f , 166yb and several Gd, Dy and Er isotopes. The 174yb and 176Hf 0 + states are discussed in terms of the pairing phase transition and in terms of Nilsson orbitals with unequal (p, t) reaction amplitudes. Members of gamma and octupole vibrational bands were observed in the even-N nuclei. The lowest L = 0 transfers to states in 169,171yb were found to have less than 55% of the strength to ground states in adjacent even-N nuclei. A strong L = 0 transfer to a state at 1513 keV in 171yb indicates the presence of a possible K = 0 core vibration coupled to the unpaired ~}[512] neutron. The natural targets have furnished information on trends in cross sections for members of ground bands, gamma bandheads, 3octupole states, and strongly excited 0 + states. N U C L E A R REACTIONS 168. 170,171,172. 173. 174-, x76yb ' 178, lSOHf' Gd, Dy, Er, HI', To, W, Os, Au(p,t), 176yb(p,p'), E = 19 MeV; measured ¢r(Et, 0), Q. E
1 6 6 ' 1 6 8 ' 1 6 9 " 1 7 ° ' 1 7 1 ' 1 7 2 " 1 7 4 y b , 1 7 6 " 1 7 8 H f d e d u c e d levels, J , 917. I 1 2 , 1 1 3 , 1 1 4 , 1 1 6 C d , 154, 155, 156, 157, 1 5 8 , 1 6 0 G d ' 160, 1 6 1 , 1 6 2 , 1 6 3 , 1 6 4 D y ' 1 6 4 , 1 6 6 , 1 6 7 , 1 6 8 , 1 7 0 E r ' 1 7 6 , 1 7 7 , 1 7 9 H f ~
lSlTa ' 182.183,184, 186W, 188,1s9,190,1920s ' (p, t), E = 19 MeV; measured Q. Enriched and natural targets.
1. Introduction T h e (p, t) r e a c t i o n o n d e f o r m e d a n d t r a n s i t i o n a l nuclei in t h e r a r e e a r t h r e g i o n has b e e n t h e s u b j e c t o f several r e c e n t e x p e r i m e n t a l i - 8 ) a n d t h e o r e t i c a l 9 - 1 3 ) studies. T h e m a j o r p o r t i o n o f t h e (p, t) r e a c t i o n s t r e n g t h f o r d o u b l y e v e n nuclei t y p i c a l l y g o e s to m e m b e r s o f t h e g r o u n d state r o t a t i o n a l b a n d w i t h o t h e r states b e i n g p o p u l a t e d w e a k l y . I n t h e S m nuclei, h o w e v e r , e x c i t e d 0 + states a s s o c i a t e d w i t h t h e c h a n g e in e q u i l i b r i u m d e f o r m a t i o n n e a r N ( n e u t r o n n u m b e r ) -- 88 h a v e b e e n o b s e r v e d w i t h o v e r 57 % o f t h e g r o u n d state s t r e n g t h 1). I n t h e G d , D y a n d E r nuclei 0 + states w i t h a p p r o x i m a t e l y 15 % o f t h e g r o u n d s t a t e s t r e n g t h h a v e b e e n r e p o r t e d 4 - 6 ). S u c h states m i g h t be a s s o c i a t e d w i t h g a p s in t h e single-particle s p e c t r u m 2) o r w i t h t Supported in part by the National Aeronautics and Space Administration and the National Science Foundation. Present address: Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540. t* Supported in part by the US Atomic Energy Commission. This is report no. COO-1265-131. 221 October 1 9 7 3
222
M . A . OOTItOUDT AND N. M. HINTZ
a nonuniform distribution of Nilsson orbitals with oblate and prolate shapes near the Fermi surface as in the actinide region 14). The present work was undertaken to investigate the (p, t) reaction on Yb and H f isotopes. These nuclei have large (fl ~ 0.3) stable deformations 15) and have an approximately uniform distribution of oblate and prolate orbitals near the Fermi surface. As we have previously reported 2), strong transitions to excited 0 + states have been found which may possibly be attributed to a local gap in the Nilsson TABLE 1 Composition of enriched Yb and H f targets a)
Target
i76yb i74yb IVayb i72yb iVlyb iT°yb 168yb
Isotopic abundance (%) 176yb
t74yb
i7ayb
i72yb
17iyb
170yb
i68yb
96.43 0.57 0.52 0.28 0.20 0.73 6.18
2.12 95.8 10.2 2.02 0.91 3.1 20.22
0.56 2.20 85.1 3.46 0.63 2.3 12.30
0.52 0.99 3.40 91.5 2.03 4.8 19.85
0.32 0.41 0.66 2.57 95.96 7.8 17.27
0.05 0.05 0.16 0.15 0.27 81.4 5.89
< < < < < <
Target
180Hf 178ttf
0.01 0.01 0.01 0.01 0.02 0.07 18.25
Isotopic abundance (%) lsoHf
i79Hf
178Hf
i77Hf
i76H' f
i74Hf
98.21 3.07
0.80 2.90
0.65 89.14
0.27 4.36
0.07 0.52
< 0.03 < 0.05
") As measured by the supplier, Isotopes Division of Oak Ridge National Laboratory.
energy level spectrum. In this work we discuss in more detail the L = 0 transitions in the H f and Yb isotopes and report results for population of 7-vibrations, octupole vibrations and other states in both e v e n - a n d o d d - N nuclei. In sect. 2 we present the experimental procedure. In sect. 3 proton scattering data for 176yb and the results of an optical-model analysis are given. Data for the (p, t) reaction on isotopic targets of 178Hf' ~80Hf and all the stable Yb isotopes are given in sect. 4. [Tabulated cross section data for the (p, p) scattering and (p, t) on isotopic targets may be found in ref. 8) or obtained from the first author (M.A.O.). ] We also present the results of a survey done with the (p, t) reaction on natural Gd, Dy, Er, Hf, W, Ta, Os and Au targets. In sect. 5 we present the results of a DWBA analysis of the t 76yb(p ' t) data and compare to recent theoretical analyses 9=13).
2. Experimentalprocedure A 19 MeV proton beam was obtained from the Williams Laboratory Tandem Van de Graaff Accelerator. Defining slits at the image of the analyzing magnet were set to give a 5 keV spread in beam energy. Data were taken using an Enge split-pole
223
(p, t) REACTIONS
TABLE2 The L = 0 cross sections and Q-values for (p, t) on natural targets Other experiments
This experiment Target
a(27½ °) (/zb/sr) ")
Qg.s. (MeV) b)
S2n c) (MeV)
Szn (MeV) ~)
Ref.
1 1 1 3
--7.8914--7.4564--7.1064--6.3634-
5 5 5 5
16.373 15.938 15.588 14.845
16.375415.938___ 15.586414.837±
3 3 3 3
71) 71) 71) 71)
154Gd 155Gd 156Gd 157Gd 158Gd 16°Gd
~ 4064-41 <~ 2784- 16 ~ 7134- 18 <~ 4554- 16 7144- 16 6164- 16
--6.6604--6.8564--6.4904--6.4174--5.8184--4.9194-
5 5 5 5 5 5
15.142 15.338 14.972 14,899 14.300 13.401
15.147± 15.339~_ 14.976414.898414.300± 13.4014-
2 8 1 2 1 2
69) 69)
~6°Dy 161Dy 16ZDy X~3Dy 164Dy
~< 8584~< 563zk ~< 859zk 51947704-
--6.9244--6.5464--6.1684--5.985zk --5.450±
5 5 5 5 5
15.406 15.028 14.650 14.467 13.932
15.415± 15.046414.652414.471 ± 13.9324-
2 7 1 1 1
164Er 166Er 167Er a68Er 17°Er
~ 501 4-125 ~< 580± 13 ~ 3894- 13 ~ 5804- 51 7504- 23
(15.744) 15.123 14.909 14.205 13.267
15.751 ± 1 15.128 zk 1 14.9044- 4 14.2094- 1 13.2624- 2
6~) 69) 69)
liZCd H3Cd 11*Cd llrCd
~ 56:k <~ 324~< 644~< 75=k
83 23 24 19 22
'.
(--7.2624-10) --6.6414- 5 --6.4274- 6 --5.7234- 6 --4.785:t- 5
69) 69) 69) 69)
69) 69)
69) 69) 69)
69) 69)
t6Syb 17°yb lVXYb 17ZYb 173Yb 174yb 176yb
3214532420745574263454544924-
16 19 10 11 8 11 10
--7.6474--6.861± --6.5994--6.1614--5.913=E --5.3594--4.2164-
7 6 5 5 5 5 5
16.129 15.343 15.081 14.643 14.395 13.841 12.698
16.0744-141 15.329± 17 15.0884- 13 14.6454- 8 14.3964- 4 13.8434- 4 12.695± 3
7o) zg) zg) z9) 67)
176Hf 177H.f 178[~.[f 179Hf lS°Hf
<~ 4354<~-3424464423045014-
34 14 15 15 25
--6.3974--6.0714--5.5314--5.2494--5.0114-
5 5 5 5 5
14.879 14.553 14.013 13.731 13.493
14.9264- 59 14.421 4- 42 14.0124- 5 13.7314- 3 13.4924- 3
zg) z9) z9) z9)
--5.7384- 5
14.220
14.2184- 37
7o)
(--6.2614-10) (--5.8104-10) --5.1244- 5 --4.4744- 5
(14.743) (14.292) 13.606 12.956
14.8954- 49 14.2434- 33 13.6064- 3 12.9544- 3
70) vo) 6s) 6s)
14.284 13.913 13.716 13.317
14.301-4- 10 13.9194- 3 13.715± 3 13.3184- 3
6s)
lSlTa
~, 354
182W 183W ls4W 186W
~ 5354- 32 ~ 164± 27 8234- 17 10504- 20
188Os
~ 6844~< 409477147784-
lS9Os 19°Os
192OS
26 18 19 16
--5.802± --5.4314--5.2344--4.8354-
5 5 5 5
2) Except for the Cd isotopes which are on a separate relative scale. b) Errors in keV. c) Errors are the same as for the Q-values.
67) 67)
67)
68) 68) 68)
224
M . A . OOTHOUDT AND N. M. HINTZ
spectrometer 16). Beam collimators were used at the entrance to the scattering chamber to give a 2 mm high by 1 mm wide beam spot on target. The (p, t) data were taken using at 3 ° (1.855 msr) entrance aperture to the spectrometer. A 2 ° (0.861 msr) and a ½° (0.211 msr) aperture were used for the (p, p) data. The Yb, Gd, Dy and Er targets were made by reducing the oxides with Ta powder in a resistively heated Ta boat and evaporating onto carbon foils 20 to 50/~gfcm z thick. The W and Ta targets were evaporated with an electron beam system. The Hf and Os targets were supplied by the Los Alamos Scientific Laboratory. Table 1 lists the composition of the isotopic materials as given by the supplier. No attempt was made to measure these compositions, but the strengths of peaks from contaminant isotopes were consistent with the listed compositions and the relative (p, t) cross sections measured with natural targets. For natural targets the natural abundances of the isotopes compiled by Fuller 17) were used. The isotopic targets contained 100 to 200 #g/cm 2 of the isotope of interest with the exception of a 440 pg/cm 2 171yb target and a 13/~g/cm 2 16Syb target. Absolute thicknesses for the isotopic H f and Yb targets were obtained using relative (p, t) cross sections from natural targets and (p, p) scattering on isotopic targets. No attempt was made to measure absolute thicknesses for most of the natural targets used. Tritons were collected using 50 #m Kodak NTB nuclear emulsion plates and position sensitive detectors (PSD)* in the focal plane of the spectrometer. Average resolution was 10 to 12 keV FWHM. Plates and detectors were calibrated by changing the magnetic field of the spectrometer to step a known peak (usually the ~76yb(p, t) ground state transition) across the detecting surface. Polyethylene foils 0.15/~m thick were used in front of the plates to stop singly charged e-particles observed at the same radius of curvature as the tritons. The plates were manually scanned in ¼ mm strips. The PSD's were 500 to 1000 pm thick with a nominal length of 3 cm and height of 1 cm. The 500 #m detectors stopped the highest energy tritons observed. In order to obtain complete excitation energy spectra, it was necessary to take runs at two or three different spectrometer magnetic fields to fill in the gaps between detectors. The signals from the PSD's were amplified and presented to the inputs of ADC's. Outputs from the ADC's were analyzed by an on-line computer using routines written a this laboratory as). Accurate ground State Q-values for the Yb(p, t) reactions were found by matching radii of curvature of the outgoing tritons. For the other isotopes studied, Q-values were found from measurements of relative excitation energies on plates. Further details may be found in ref. 8). The ground state Q-values and (p, t) cross sections for the lowestL = 0 transfers are given in table 2. The contents of table 2 are discussed in more detail in subsect. 4.4. t Supplied by Nuclear Diodes, Inc.
(p, t) R E A C T I O N S
225
3., Elastic and inelastic proton scattering To get absolute cross sections and proton optical-model parameters for the Yb(p, t) reactions, data on the lv6yb(p, p) reaction was taken. The 2 + and 4 + members of the ground state rotational band were also observed for 0~ab > 22½° and 35 ° respectively. Fig. 1 shows a typical spectrum. Carbon, oxygen and 4°Ca 176 YB (p,p) 176yB d
2+0
Ep =19.0 M e V
400 z < "r" (.3 5 0 0 r'~ ILl t3... 2 0 0 I"== 100
0+0
85°
0 L =
,4%
4+0
o
.......
0
...&, ............ I00
50
,J 200
150 CHANNEl
250
Fig. 1. Position sensitive detector s p e c t r u m for 176yb(p, p) to the first three m e m b e r s o f the g r o u n d state rotational band. Lines d r a w n to guide the eye.
I o_
b ~-
IzsYB ( p,p)lZSyB
o.2 0.1
5d
,
2O ~.
0+0 *
,
,
IO
03
"
"~
*
,
5
2+0
.
.El
°
E ~=
2 I
,~ b~
0.5
"Ol'o
0.2
°
.
~
....
".
t" ÷
;, {,,. ]
4+0
". °
I
0
20
~
,40
I
60
I
80
°
*
°
I
mo
°
°
•
*
I
~20
8cm
Fig. 2. A n g u l a r distributions for the first three m e m b e r s o f the g r o u n d state b a n d o f 17eyb excited by p r o t o n scattering. Values o f J~rK are indicated. Relative errors are s h o w n where they are larger thart t h e d a t a points. T h e absolute n o r m a l i z a t i o n error is a b o u t 10%. T h e solid line is the opticalm o d e l fit to the elastic scattering data u s i n g p a r a m e t e r set 5 o f table 3.
contaminants were identified by kinematics. Corrections were made for the presence of other Yb isotopes in the target by assuming the cross sections to members of ground bands for these "contaminants" are equal to those of 176yb" A partial justification for this assumption may be found in the 16 MeV proton scattering done by Kruse e t al. ~ 9) where it was found that the cross sections for 0 + and 2 + members of ground
226
M.A. OOTItOUDT AND N. M. HINTZ
bands were not greatly different for 172yb ' ~7 4 y b and 176yb" The angular distributions are plotted in fig. 2. Error bars are shown where they are larger t h a n the data points. The elastic scattering data was fit by the optical model [parameterized as in ref. zo)] with the automatic search routine R A R O M P zl). The C o u l o m b radius was taken f r o m a fit to electron scattering data 22) and spin-orbit parameters were taken f r o m the global parameters o f Becchetti and Greenlees 20). In order to avoid overTABLE3 Optical-model parameters for 176yb(p, p), E = 19 MeV Set
0 c) 1 2 3 4 5
Real
Imaginary
V (MeV)
r ") (fm)
a (fm)
57.824 56.4588 56.242 54.2057 55.6437 54.2725
1.17 1.202 1.2005 1.2306 1.2055 1.2315
0.75 0.665 0.642 0.6673 0.6587 0.6614
Wv (MeV)
WD (MeV)
r a) (fm)
Spin-orbit a (fm)
1.4811 9.5033 1 . 3 2 0.653 0.0623 10.8819 1.0614 1.0592 0.0006 10.6395 1.0929 1.110 11.0035 1.0754 1.0281 7.6155 1.4776 0.9509 12.5116 1.0253 1.0206
Z2 b)
V r a) a (MeV) (fro)(fm) 6.2 6.2 6.2 6.2
1.01 0.75 204 1.0 1.01 0.75 1.3 0.9 1.01 0.75 1.6 1.01 0.75 1.1
") Multiply by AT. ~) Chi-square per point, assuming a 3% error for all data points. c) Parameters from ref. 20). emphasizing the forward angles where the Y b first excited state and 4°Ca were n o t resolved f r o m the g r o u n d state, a standard error o f 3 % (approximately double the statistical error for most points) was used for all points. A normalization grid was performed to give the absolute cross section with an estimated 10 % error. Five parameter sets arrived at are shown in table 3 along with the Becchetti-Greenlees 20) global set. A t the conclusion o f the p r o t o n scattering run, the g r o u n d state transition in 176yb(p, t) was observed at 01,b = 27½ ° near the L = 0 maximum. Using the absolute normalization f r o m the p r o t o n scattering, the absolute cross section for the g r o u n d state transition is 492 pb/sr _ 10 %. D a t a for elastic p r o t o n scattering on 178Hf were taken f r o m 17½° t h r o u g h 27½ ° in 2½° steps. The cross section was assumed to have the same ratio to Rutherford as 176yb(p ' p). The absolute cross section for the 178Hf(p, t) ground state transition at 01,b = 27½ ° was f o u n d to be 464 ktb/sr + 20 %.
4. The (p, t) data and discussion Recent experimental a, 5 - 7 ) and theoretical 10, 12) investigations have shown that for deformed nuclei, the shapes o f (p, t) angular distributions for a given L-value are quite variable. (See also fig. 3,) D u e to the great length o f time necessary to
(P,T) ANGULAR
DISTRIBUTIONS L = 2 TRANSFERS
L=O TRANSFERS i l
t
t 174yB 0 keY : Q:-4.216 MeV
I
i
I
""
174yB, 76 keY Q=-4.292 MeV
q
] !
~,
I
172yB,78 t.',eV Q=-5.457 MeV
]~"N t
/ I
! \I/'
Q°-5.359MeV
I
,/ :' V: ~,174yB,1489 keY I
J\Mev I
E b
I
--~
b~
'~174yB,1886 keY ~-6.I02MeV
/ ~
I
4yB
'
i
i
168yB,87 keY Q=-6.948 MeV
I
~
i
~
1656 Re',
'
,
,
i
I
]
I I
I ~.L... , 168yB.984 key L "x...L.//I "% Q=-7.845 MeV 1174yB,2100 keV
.~
S
~
f X " Q=-6.3t6 MeV i
~
i
eV
,L°4! T,RANSFERS
t
i
I
"~@-.. ', ',
L l
t
I
>,,,"--T'~ , i 174yB,254 keY /I ~ ~"';,,,,_~Q=- 4.470 MeV
,
M
:
I I
' 168yB 0 keY
k
i
i
]
~
[
t
I
~, 168yB,284 keV Q=-ZI45 MeV
i
,
i
t
I
20
40
I
i
60 8c.m.
I
I
80
I
I
I
0
I0 0
[
20
I
40
60 ec.m.
80
I
I00
Fig. 3. Shapes for (!0, t) angular distributions labelled by excitation energy and Q-value. The cur are s m o o t h lines drawn t h r o u g h the data points. F o r each value of L, the curves are arranged vertic~ in order o f decreasing Q-value. These curves were used in C U R V E F I T as discussed in sect.
250O [L.
i
'
I
EXCITATION ENERGY (keV) 2000 1500 I000
'
174yB (p,f)172 YB Ep = 19MeV eL= 271/2°
~_ 400
200
500 0 6+ G.S. BANDo+ ] ~ I
2+ 0 +
Or)
")÷
¢o I00
2+ 0 ~~ f
t-0
-~r~
I/©
115 120 125 DISTANCE ALONG PLATE (cm)
150
Fig. 4. Triton spectrum for the t 7 4 y b ( p , t)172yb reaction. The K = 0 bands are indicated. C o n t a m i n a n t peaks are labelled " C " .
228
M.A. OOTHOUDT AND N. M. ItlNTZ
178HF, 0 keY O+OA
176HF, 0 keV O+OA
168yB,II98 keY 0 + 0 B
172yB, 1040 keY
168yB, 1340 keV (O+OC)
172yb,179l keV 52 keY O+OB
~90+Oc 168yB, 1543 keV O+OD
2 keV 178HF,1448 ReV
11~~'9
'~
2I " ~+yB,+°° 2504 keV
keY
/~ O+OB ,
3
f / ~ 178HF,1776 keV
~
0+0C
170yB, 0 NeV
E v
168yB, t904 key (O+OE)
(O+OE)
7B~
O+Oc
HF, 2024 keY O+OD
I HF, 1749 keV i O+OD
168YB, 2t60 heV (O+OF)
/1/ ~ !
174yB 2821keY
J
/
166yB,
/ 0 ,eV O+OA
IZa~F, 2316
keV
~(O+OEIl
I
I
1
0 20 40 60
i,
,~
¥~3, 2904 keV / ~ (04-0F)
I
I
I
0 20 40 60 0 20 40 60 Ocm
P //~ 166yB, 1045 keV (O+OB) f
0 20 40 60
Fig. 5. Angular distributions for 0 + states not shown in fig. 3. The residual nucleus and excitation energy are indicated. Curves are labelled by ]~K followed by a letter to distinguish bands. Lines are drawn to guide the eye. compute angular distributions with coupled-channel computer codes and to a lack of proper wave functions, L-values were identified by comparison of the data with the shapes of angular distributions for transfers to known states. Assignments made onthe basis of this phenomenological procedure are considered to be tentative unless
(p, t) REACTIONS
229
they agree with assignments made in other experiments or with I(I+ 1) spacings in proposed rotational bands. Identification of L-values by shapes of angular distributions was done with the computer code C U R V E F I T 23). A set of standard shapes (shown in fig. 3) for known L-transfers was read into the Williams Laboratory CDC 3100 computer. The code normalized the standard curves to an unknown curve to give the best chi-square per degree of freedom (X2), where the number of degrees of freedom equals the number of data points less the number of adjustable parameters. Here the only parameter is the normalization. The fits were ordered from best to worst Z: and displayed on an oscilloscope screen. Provision was also made for adding together two reference curves to obtain the best Zz. Angular distributions were taken on states to beyond 2.5 MeV of excitation energy in the H f a n d Yb isotopes (2.2 MeV in 166yb). A typicalspectrum is shown in fig. 4. For the Yb isotopes, searches were made up to 4 MeV of excitation at 27½° near the L = 0 maximum. No states were found beyond 2.5 MeV with strength greater than 5 ~ of the ground state transition. ( F o r ~66yb the search was to 3.3 MeV with no states stronger than 10 ~o of the ground state being found beyond 2.2 MeV.) Extensive angular distributions are shown in fig. 3 for states that have been assigned 0 + and for known 2 + and 4 + states. The curves are arranged vertically in order of Q-value. The L = 0 transfers have a very distinctive oscillatory pattern that is damped out as the Q-value becomes more negative. However, the shape is well defined by the cross sections at the laboratory angles indicated by the dashed vertical lines: 12½°, 27½°, 42½° and 55 °. The L = 2 shapes are quite variable, although the 7-vibration bandheads (third and fifth curves from the top) are similar to each other and the ground state band 2 + states (the remaining curves) are similar to each other. Only two extensive L = 4 angular distributions were taken. They do not resemble each other (especially at low angles) and the ~6syb 4 + is somewhat similar to the L = 2 distributions. Angular distributions for the other known L-transfers were not sufficiently characteristic to assign L-values. For most states data were taken only at the four angles identifying L = 0 transfers. These four angles will be called the "standard angles." The summed cross sections given in this paper are sums over these angles without a sin 0 factor. Unless otherwise noted, the percentage figures given to indicate the strength of excited states are the ratios of the summed cross sections for the state to the summed cross section for the lowest strong L = 0 state (the ground state for even-N targets). Angular distributions over the standard angles for 0 + states are shown in fig. 5. 4.1. SELECTION RULES The initial and final nuclear total angular momenta and parities are denoted Ji, Jr, zq and ~f. The total orbital and spin angular momenta of the transferred pair are designated L and S. Assuming the two neutrons in the triton are in an S -- 0 state with relative s-state motion [which has been calculated 24) to be 90 ~ of the
M. A. OOTHOUDT A N D N. M. HINTZ
230
TABLE4a States in 'TSHf Other experiments
This experiment J~rK (keV) 0 92 304 635 1088 1179 1272 1325 1387 1448 1510 1562 1643 1776 1816 1874 1947
(pb/sr)
0 10 10 10 10b) 10 10 10 10 c) I0 10 a) 10 10 10 10 a) 10 10
862.7 220.3 68.5 9.0 1.3 39.3 3.3 41.2 11.4 23.4 14.4 7.4 4.5 42.3 39.3 23.3 11.3
9.7 6.9 4.1 1.0 1.1 5.3 1.1 2.8 1.6 1.6 1.6 1.1 0.7 3.0 3.1 2.0 1.2
0+0A 2+0A 4+0A 6+0A 2+2
4+2 0+0B 2÷0B
0+0C (2+0C)
Ex (keY)
93 306 632
yTrK
Ref.
2+0 4+0 6+0
7s) 7s) 30)
1175 1269 1323 1384 1444 1515 1562 1636 1772 1819
2+2 3+2 3-2 4+2 0+0 2+(0) 2+(0) 4+0 0+0 2+0
3o) 3o) zo) 3o) 30) 3o) 30) 3o) 7s) 3o)
1954
(4)+(3)
30)
Other experiments
This experilnent Ex (keY) 1991 2024 2056 2121 2156 2203 2227 2286 2316 2371 2393 2435 2474 2572 2628 2668 2707
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
Xae.m. ~) ~b/sr) 11.1 42.3 10.1 14,5 10.5 20.8 23.9 19.2 44.2 42.0 18.5 14.2 16.8 6.8 6.3 14.1 6.9
1.2 2.1 1.2 1,6 1.4 2.0 2.1 2.1 3.1 3.2 2.1 1.6 1.5 1.3 1.1 2.0 1.3
.l n K
Ex (keV)
ar~rK
ReL
0+0D (2+0D)
(0+0E) (2+0E)
(2 +)
TABLE 4b States in xe6Hf Other experiments
This experiment ~K (keY) 0 87 287 600 1152 1231 1293 1314 1343 1362 1387 1511 1545 1607 1678
0 10 10 10 10 10 10 10 10 10 10 10 10 10 10
(/zb/sr) 812.7 15.3 219.2 8.8 44.1 4.3 8.3 1.5 95.0 4.6 28.7 3.0 63.8 4.6 32.0 3.1 55.5 4.3 6.7 2.5 19.2 2.5 5.6 1.1 15.4 1.9 3.7 2.1 15.3 1.8
0+0A 2+0A 4+0A 6+0A 0+0B 2+0B 0+0C 2+2 (2+0C)
E,~ (keV)
.17rK Ref.
88 290 597 1150 1227 1293 1313 1341
2+0 4+0 6+0 0+0 2+0 0+0 3-2 2+2
3s) as) as) as) 35) 35) 3~) 35)
1379
2+0
35)
1678
4+3
52)
(4+2)
Other experiments
This experiment Ex (keV) 1749 1796 1857 t953 2049 2069 2089 2136 2286 2304 2348 2389 2415 2448
10 10 10 10 10 10 10 10 10 10 10 a) 10 ~) 10 10
~ c . m . ") (~b/sr) 41.9 14.7 22.3 5.9 17.4 12.0 10.7 18.4 22.7 6.9 20.5 25.4 14.1 15.1
3.6 2.2 2.6 2.3 2.3 1.8 1.6 2.6 2.7 3.0 1.9 2.4 1.8 1.8
ar~K
0+0D (2+0D)
Ex (keY)
J~rK
1794 1862 t957 2049 2065
(3-) 2+ 2-2 1+ 1 2+(1)
2281 2307
(1-) 1+
Ref.
3s)
79) 36) a6) 36)
3s) 36)
TABLE 4C States in 174Yb This experiment Ex (keY)
0 76 254 526 895 1316 1386 1489 1520 1564 1608 1636 1676 1712 1783 1807 1852 1886 1959 2066 2100
~ . . . . ~) (/zb/sr)
0 5 5 5 5 5 5 5 5 5 10 5 10 5d) 5 5 5 ¢) 5 5 10~) 5
818.6 237.4 62.5 15.1 2.2 2.8 38.0 210.5 7.7 66.9 4.6 43.3 5.5 39.0 4.9 13.2 25,3 96.9 46.9 11.9 59.2
4.9 2.2 0.9 0,7 0.4 0.3 1.9 2.9 0.6 1.2 0.4 1.3 0.5 1.2 0.7 0.8 1.3 1.5 2.3 0.8 1.7
Other experiments JrrK
0+0A 2+0A 4+0A 6+0A 8+0A
,FrK
Ref.
76 253 526 892 1318 1380
2+0 4+0 6+0 8+0 2-2 3-2
47) 8o) so) 8o) 47) av)
1518 1559
(6 + ) 5-2
47) zv)
1630 1667 1702
2+2 6-2 3+2
37) 37) 37)
1800 1846 1884
4+2 3(5-)
3v) aT) 47)
Ex (keY)
0+0B 2+0B 2+2 4+0B 4+2 0+0C 2+0C o+oD
This experiment Ex (keV) 2127 2174 2246 2300 2342 2375 2436 2469 2520 2558 2588 2620 2662 2720 2753 2821 2840 2882 2904 3004 3042
5 5 d) 10 5 10 5~) 5 5 10 5 5 5 5 5 5 5 10 5 5 5 5
Other experiments
Y'~. . . . a) (ub/sr)
j~rK
13.4 0.8 55.1 2.4 9.0 1.0 4.8 0.6 12.3 0.9 7.3 1.3 12.7 0.9 19.0 1.1 6.2 0.9 50.2 2.4 31.6 1.4 10.5 1.2 29.6 1.6 11.9 1.2 9.0 1.2 12.0 1.5 21.3 1.6 7.6 1,0 14.5 1.3 24.2 1.6 6.7 1.4
4+0C 2+0D
4+0D
Ex (keV)
J~rK
Ref.
2184
2+2
av)
2333 2370
4+2 4+3
37)
a7)
(2 + )
(0+0E)
(0+0F)
TABLE 4d States in 172yb This experiment E~ (keY)
0 78 258 539 911 1040 1115 1154 1171 1218 1260 1281 1350 1410 1467 1508 1536 1556 1604 1628 1656 1705 1757 1791 1819
E ~ . . . . a) (/zb/sr)
0 886.3 11.1 5 287.6 6.3 5 80.1 2.3 5 15.9 1.1 I0 1.6 1.1 5 37.0 1.6 5 29.6 1.5 5 ¢) 14.1 1.0 5 8.6 0.9 5 18.2 1.2 5 c,~) 48.9 2.0 5 7.5 1.0 5 8.9 0.8 5 4.8 0.7 5 a) 68.1 2.2 5 10.2 1.0 10 6,7 0.7 10 7.5 0.8 5 7.0 0.9 10 4.1 0.7 5 d) 24.5 1.9 5 18.9 1.6 5 4.2 0.8 5 25.6 1.5 5 32.0 2.2
Other experiments d~rK
0+0A 2+0A 4+0A 6+0A 8+0A 0+0B 2+0B
4+0B
2+2A
2+2B 4+2A
0+0C
E~ (keV)
79 260 540 912 1043 1119 1155 1172 1222 1262 1283 1352 1405 1466 1510 1541 1549 1609 1632 1658 1708 1749 1794 1822
drrK
2+0 4+0 6+0 8+0 0+0 2+0 1-1 3+3 3-1 4+3 4+0 5-1 0+0 2+2 6+3 6-1 3+2 2+2 4+0 4+2 3-0 4+3 0+0 3-2
Ref.
a2) 8o) so) so) 32) a2) 48) s*) 51) 81) av) 51) a2) 81) 81) 51) 81) 8*) 48) 81) st) 81) az) 48)
This experiment E~ (keV) 1846 5 1892 5 1916 5 1954 5 2004 7 2025 7 2041 7 2060 7 ~'e) 2098 7 2184 7 2224 7 2254 12 2337 7e) 2364 7~) 2396 12 2436 7 2460 7 2483 7 c) 2540 10 ¢) 2580 10 2734 10 2786 10 2817 10 2832 10
Other experiments
Y,Oc.~. a) (/zb/sr)
,FrK
Ex (keV)
ylrK
Ref.
13.3 1.4 22.4 1.8 14.0 1.0 17.5 1.1 6.7 1.0 4.5 0.8 8.2 1.1 9.2 0.9 3.2 0.6 5.3 0.9 25.6 1.3 5.4 0.7 32.9 1.6 15.8 1.4 4.8 1.3 10.1 1.1 11.0 1.1 32.0 2.8 30.7 2.3 13.3 1.8 6.8 1.6 6.3 1.4 9,2 1.9 8.7 1.8
2+0C 0+0D
1850 1894
2+0 0+0
32) 32)
2+0D
1956 2009 2032 2047
2+0 1+t 32+1
z2) 81) 81) st)
2344
4+
81)
2480
(0) +
48)
2741 2786
9-5 8+8
sl) sl)
(2 +) (2 +)
(2 +) (0+0E)
232
M . A . OOTHOUDT A N D N. M. t t l N T Z TABLE 4e States in t 7 ° y b
This experiment E~ (keV)
Y'a¢.m. ") (~b/sr)
0 0 85 5 277 5 572 5 1072 5 1144 5 d)
904.2 323.9 77.0 11.3 15.6 72.2
9.8 6.6 3.1 1.2 1.4 2.3
1225 10 1331 5
8.1 24.4
1.0 1.8
1350 1360 1398 1478 1513 1534 1568 1602 1634 1657 1711
8.9 8.3 23.5 13.4 8.7 8.8 27.9 3.1 10.1 4.2 7.6
1.1 0.8 1.2 1.3 1.0 1.1 2.0 1.4 1.2 1.2 1.0
5 5 5 a) 5 5 5 5) 5 5 5 5 5
Other experiments J~K
E~
JrrK
Ref.
(keV) 0+0A 2+0A 4+0A 6+0A 0+0B 2+2, 2+0B 4+2, 4+0B
0+0C 2+0C
(4+0C)
This experiment Ex (keV)
84 278 573 1069 1138
2+0 4+0 6+0 0+0 2+2
as) s3) a7) a3) a3)
1780 1838 1871 1971 2001 2047
1222 1325
3+2 4+2
a7) 37)
2137 12 2171 7
1360 1400 1480 1512 1534 1566
1-0 30+0 1-0 2+0 0+0
3a) sT) 3a) aa) as) as)
1634 1651 1714
2+0 2+0 2+1
2229 2249 2281 2325 2352 2390 2437 2492 3a) 2539 54) 2595 54) 2678
5 c) 5 5 10 10 c) 7
7 7 7 7 12 7 12 7 7 7 7
~--,t~c.m.")
Other experiments
J~rK
(/~b/sr) 27.8 7.4 4.9 12.0 8.6 16.7
1.4 0.7 0.7 1.3 0.8 1.t
9.0 1.1 21.7 1.6 14.2 8.7 12.9 29.9 7.3 9.9 16.5 8.0 20.5 5.1 6.8
E~
J~rK
Ref.
(keV) 1783 1832
32+1
37) 5*)
2040
1+ 1
33)
2275
1-0
3s)
2533
1+0
3s)
(2 + )
1.2 0.9 1.2 2.3 1.3 1.7 1.4 1.4 2.1 1.6 1.8
TABLE 4f States in 16Syb This experiment
.E,~
Y'ac.m. ~)
(keV)
(#b/sr)
0 87 284 586 886 943 984 1064 1098 1161 1168 1198 1231 1277 1340 1480 1543 1600
0 5 5 5 10b) l0 b) 5 5 5 7e) 5 5) 5 5 5 7") 5 c) 5 5a)
938.1 361.9 93.2 10.8 2.9 1.9 45.8 4.3 5.6 3.1 14.4 19.7 7.8 20.1 2.8 24.2 26.5 47.0
11.4 6.9 3.9 0.7 1.1 1.1 1.7 0.5 0.8 1.1 1.0 1.2 0.5 1.6 0.6 1.2 1.9 2.3
Other experiments
JrrK
E,,
JrrK
Ref.
(keY) 0+0A 2+0A 4+0A 6+0A
2+2
4+2 0+0B 2+0B (0+0C) (4+0B) 0+0D (2+0D)
88 287 585
2+0 4+0 6+0
s*) 34) 34)
984 1067
2+2 3+2
s4) a*)
1156 1171 1197 1234
0+0 4+2 0+0 2+0
34) 34) 82) s4)
1475 1543 1595
30+0 3-
37) s2) 37)
This experiment E~ (keV) 1698 1727 1767 1793 1859 1904 1973 1993 2092 2121 2160 2173 2186 2292 2327 2360 2464 2500
5 5 5 5 5 °) 5 5 5 7 7 7 12 7 7 7 12 12 12
Z(rc.m" a) (#b/sO 10.7 5.7 6.2 6.7 16.7 37.4 9.0 30.1 5.8 5.2 12.2 6.4 7.3 7.7 6.9 6.4 8.8 4.7
0.9 0.7 0.7 0.6 1.1 1.7 1.1 2.0 1.1 1.2 1.2 1.5 1.2 1.1 1.0 1.2 1.5 1.5
Other experiments
jz, K
Ex
JZrK
Ref.
(keV)
(4+0D) 1770
(0+0E) (2+0E)
(0+0F)
5-
a*)
(p, t) REACTIONS
233
TABLE 4g States in 166yb Other experiments
This experiment Ex (keV) 0 0 101 10 329 10 a) b) ~) ~) D
~r~.m. a) (#b/sr)
yzrK
646.9 22.0 267.2 16.1 54.8 8.0
0+0A 2+0A 4+0A
Ex (keV)
102 330
JrrK
2+0 4+0
Ref.
s3) 8a)
This experiment Ex (keV) 931 10 1043 10 1581 10
Z~r.... a) ~b/sr) 57.7 7.8 75.9 11.0 29.8 15.5
Other experiments JZrK
Ex (keV)
Jr:K
2+
(0+0B)
Sum of center-of-mass cross sections over standard angles. Unidentified contaminant? Possible doublet. Doublet. Contaminant subtracted.
time], it has been shown 2 5) that the following approximate selection rules hold:
IJi-Jel < L < Ji+Jf, = ( - 1 ) L.
(1)
(2)
For deformed nuclei further selection rules may be found. Let K i and Ke be the projections on the symmetry (z) axis of the total initial and final angular momentum in the intrinsic frame. Sufficient angular momentum must be supplied to satisfy eq. (1) and to change the projection on the z-axis; thus t __> [gf-K~[.
(3)
The Z and A selection rules result from consideration of the projections on the z-axis of the spin and orbital angular momenta of the transferred neutrons 26). The neutrons are in Nilsson levels labelled by the asymptotic quantum numbers £2[NnA]S, where f2, A and 27 are the projections of the total, orbital and spin angular momenta of the orbital on the z-axis, N is the total number of harmonic oscillator quanta and n is the number of quanta on the z-axis. The parity of the orbital is ( - 1)N. A pair of particles may be picked up with the projections of their individual total angular momenta parallel ( + sign) or antiparallel ( - sign):
01 [N1 n, A1 ]271+-f2z [N2nz A2 ]S=.
(4)
The projection of the total angular momentum of the pair on the z-axis is K = 12, +122 . If we assume as before S = 0, then we must have 271+2:2 = ~ -
& = 0.
(5)
Further we note that L must be large enough to give the required change in the A components, i.e., Z > [hl+_hzl, (6)
Ref.
234
M . A . O O T H O U D T A N D N. M. H I N T Z
where the sign is chosen in agreement with eq. (4). Eqs. (5) and (6) will be referred to jointly as the 2J-A rule.
I80H F(p,f) 178HF (0 +)
0+
0+
I1
I OOC
(5+)
Fp
0+
0 +
I
I-12~
o+111
I
I
I
:::k
0+
~z2-
(2 + )
I00
I
v
I
v
b k'q
i
,
i
j
i
i
i
2.0
2.5
1.5
1.0
0,5
0
EX (MeV)
Fig. 6. Schematic spectrum for 18 o E f ( p ' t)17 SHf" F o r each state the s u m m e d cross section is plotted at the p r o p e r excitation energy. The s u m is over standards angles (01.b = 12½°, 27½°, 42½° and 55°). K n o w n rotational bands are indicated. Values o f K ~r are s h o w n for bands and J~ for some individual states. The descending vertical lines are longer if experimental candidates for the state exist. I
I
]
]
i
J
L
I
i
I'
4
I
]
J
I
~
I
J
i
J
i
178 HF(p,f)176HF
I000
--
0+ 0 ÷
0+
I~
I-] I-]
] I
(6+)
5+
,~?~l~l {+ I
--. I O0
i + I+
gb b-q
0+
2+
' 12-
~ ]
~q~ Io -
2.5
2.0
L5
1.0
0,5
0
Ex (MeV)
Fig. 7. Schematic spectrum for 17 s H f ( p ' t)~v6I_If. See caption to fig. 6.
Aside from the assumption that the neutrons in the triton are in an S = 0 state, t h e above selection rules are based on the exactness of the asymptotic quantum
(p, t) REACTIONS
235
numbers for finite ft. Rassey 27) has shown that for a finite deformation the wave function is a linear combination of asymptotic states. In the rare earth region the '
i
,
I
i
I
t
i
l
,
i
,
,
I
,
i
i
i
I
~
i
i
i
i
i
176 YB(p,t) 174yB (o +)
0+
0+
0 4-
O+ [---]-] 3- I---~
-~ I000 ..Q
2+ 5+[--~
::L
i
F-P
I00
z-
q-T]
Q:>
b
Pq
I0
Jlll,l ,II[I lI I
1 I
2.5
t
i
t
,
I
i
2.0
I
J
i
i
i
1.5
i I i i i
t,0
f t l ,
0.5
0
Ex (MeV)
Fig. 8. Schematic spectrum for 176yb(p, t)tV4Yb. See caption to fig. 6.
I74yB(p,t) 172yB 0+
FTTq
_Q
(0+) I00
q~
b I0
0 4-
0+
0+
~,- I 0 0 0
::L
2+
I+
I~ 3-
l
3-2 +
5+
I
Ilti ,:iliil,]iill I
2.5
2.0
t.5
1.0
0.5
0
Ex (MeV)
Fig. 9. Schematic spectrum for 17~yb(p, t)t72yb. See caption to fig. 6.
asymptotic label has the largest amplitude, although there are significant c o m p o n e n t s which allow relaxation of the Z - A rule. Thus 27 = 1 two-quasiparticle states m a y be p o p u l a t e d in (p, t), but m o r e weakly than I; = 0 states.
M. A. O O T H O U D T
236
A N D N. M. H I N T Z
4.2. E V E N - N Y b A N D I-If T A R G E T S
D a t a for the states observed in this experiment are given in table 4. Excitation energies and errors (keV) are given in the first two columns. The next two columns J
'
'
J
'
'
'
i
'
'
'
'
I
. . . .
J
....
' ' '
172yB(p,t) 170yB 0+ I000
0+
0 +
J
I+ I+ 0 +
FTTi ( 2 +)
I
:::L IOO
b
3"
3-
I
I
llla,JI,I,1111 I
t0
J
2.5
J
2.0
i
1.0
15
0.5
Ex (MeV)
Fig. 10. Schematic s p e c t r u m for 172yb(P, t)17°Yb. See caption to fig. 6.
170yB (p,t) 168yB (O~i (0 +)
0 +
O+
0 +
A 03
"~ 1o o o _Q ::L
•
I
'"
3- 3-(0+) I L
III
v
I00 b I0
i
2.5
20 .
1.5 Ex (MeV)
1,0
0.5
o
Fig. 11. Schematic s p e c t r u m for lV°Yb(p, t)168Yb. See caption to fig. 6.
give the sum of the cross section over standard angles and its relative error. Absolute cross section errors are approximately 10 % for Yb and 20 % for H f isotopes. As-
(p, t) REACTIONS
237
168yB(p,t)166yB (0 +)
0+
2+
I000
1
n :::L I00 b tO
t
,
i
2.5
J
[
1
~
,
i
2.0
[
B
i
i
i i i i i
i
1.5 I0 Ex (MeV)
i
0.5
0
Fig. 12. Schematic spectrum for 168yb(p, t)166yb. See caption to fig. 6. Because of the poor quality of the 168Yb target, only the strongest states could be identified.
1ooo -
1
I
dodw ~b/sr
I
i
1
I
ybt76(p,t) Yb I~ ~
I00
Ep = 19 MeV
~
O÷
2+ I0 4+
•~ 0o
F
20 °
t
40 °
I
t
60 °
80 °
I
I00 o
8CM
Fig. 13. Angular distributions for members of the ground state band of 174yb populated by 176yb(p, t). Lines are drawn to guide the eye.
238
M . A . O O T I - I O U D T A N D N. M. H I N T Z
signments for J~K made in this experiment are given in the next column. A letter after the K assignment indicates which of two or more bands with the same K-value the state belongs to. The final columns give a partial listing of excitation energies
.,oool I
/
GROUND
STATE
BANDS
"
+
P,,l
-YB AHF
I.o ~-
•
I
2+ 0.1
+
o b Pq A
4+ i
0.01
¢
-.) v
b P4 0.001
I 96
i 98
RESIDUAL
t
f I00
I02
NEUTRON
4'
•
t'
8+
i 104
r 106
6+
NUMBER
Fig. 14. S u m m e d cross sections for g r o u n d states a n d ratio to the g r o u n d state s u m for m e m b e r s o f g r o u n d state bands. T h e s u m is over s t a n d a r d angles. T h e g r o u n d state s u m s h a v e a n absolute e r r o r o f 10% a n d 2070 for Y b a n d H f respectively. Solid lines indicate values o f ( A / G ) 2 normalized to t h e data. T h e s a m e n o r m a l i z a t i o n is u s e d for I-If a n d Yb.
I "-~" ;
- 5{-
~,~2 I"- ~ 2 . 0 ~ ~_.-[ - " t Sf0 ~ t
x,,~
K=O BANDHEADS 4% 2"1"2"1"
5" O i
i.0r
5% - - 5 % I' 5%-- " 5*/. I°z~3°1:~6./~ 26% -I ~ , 5 */* 0~/~ o ~ - - - - ~ v . . _ * --,04%8% . t~-5o/< I2% <0-~*2% -c--'2°/o 4% - - 12Y. . . . , I%
7%,2%
,
~
5%
--
5%
--
Fig. 15. The E = 0 bandheads in the Y b and H £ isotopes studied. Percentages give ratios of summed, cross sections for excited 0 + states to the ground state. Dashed lines indicate 0 + states not seen in this experiment; u p p e r limits on the strengths o f these states in (p, t) are indicated.
(keV) and assignments from other experiments. "Contaminant subtracted" means that a state from anothe r isotope of the element considered has been subtracted to give the cross section indicated. In figs. 6-12 we plot the summed cross sections
(p, t) REACTIONS
239
versus excitation energy. Known bands are indicated above the schematic spectra; short lines correspond to known states not observed in (p, t). 4.2.1. Ground state K = 0 bands. Data were taken on ground state rotational bands up to the 6 + member in most cases and to the 8 + member in some. Fig. 13 shows extensive angular distributions for members of the ground state rotational band up to the 8 + member populated by 176yb(p, t)174yb. Fig. 14 shows ground state strengths and the ratios of summed cross sections in the ground band to the ground state strength. In the PWBA-BCS approximation the ground state cross section is proportional to (A[G) 2, where A is the BCS gap parameter and G is the strength of the pairing force 2s). This term is plotted in fig. 14 using G = 20/A MeV. Here and later A is set equal to the odd-even mass differences of Whineray et al. 29). Average values of G and A over the target and residual nucleus are plotted, since it is not clear which should be used. The fit is poor, reflecting the inadequacy of the PWBA. The sharp drop for the 4 + and 6 + cross sections of 176Hf with respect to 1 7 4 y b may be due to an abrupt change in the Y4o moments. From inelastic e-scattering 15), /?4 is known to change from -0.054 for 176yb to -0.072 for 178Hf. 4.2.2. Other K = 0 bands. As previously reported 2), a number of excited 0 + states are populated by the (p, t) reaction in the Yb and Hf isotopes. Angular distributions for the 0 + states are plotted in figs. 3 and 5. The proposed 0 + states are shown in fig, 15 along with 0 + states not seen in this experiment but reported by other workers 30- 34). The 0 + states were generally observed with 2 to 5 ~ of the ground state summed cross section. The population of 0 + states in 176Hf at 1152 keV (with 12 ~ of the ground state strength) and 1293 keV (8 ~), in 174yb at 1489 keV (26 9/o), 1886 keV (12 ~o), and 2100 keV (7 ~o) and in 166yb at 1043 keV (12 9/0) is therefore distinctly anomalous. The 0 + states in 176Hf have been previously observed 35, 36) in the decay of 176Ta" The strong Yb 0 + states have not been previously reported, but the excellent L = 0 shapes for the angular distributions (figs. 3 and 5) and the presence of well developed rotational bands built on these states for 174yb (table 4c) make 0 + assignments very certain. For ~66yb background was too high to observe higher band members for the 1043 state. The suggested 2 + (2174 keV) and 4 + (2342 keV) members of the 2100 keV band in ~74yb are near members of a K = 2 band observed by Burke and Elbek 37) with a 2+2 at 2184 keV and 4+2 at 2333 keV. The 2174 keV peak is very broad in the (p, t) spectrum indicating it is an unresolved doublet; it has a reasonably good L = 2 shape indicating both components of the doublet are 2 +. Several explanations have been put forward to account for the numerous 0 + states observed in the rare earth region: I (i) The /?-vibrations considered by B~s 3s) ~esult from residual quadrupole interactions which are typically written i i
-- z f ( r , , rz)Y*,(l!')Y2~(2).
240
M.A. OOTHOUDT
A N D N . M. H I N T Z
SINGLE NEUTRON
LEVELS
3[ 501] . . . . . . . . . . . . . . . . . . . . . . . . . . . /-----
[1[6153 . . . . . . . . . . . . . . . . . . . . . , i't503]
,, -71514.]-.
1,,,--
;z:
Y.
~,
/ .~.__71503] I ~. . . . 5[5 t2]t
.....
T
,~------7'[st4]j
--/" q---~l"-" ~-~- 5ts~q
---~,".--'~/
",~.~,L
,~,o~-:-'-~,':--,"~/
,=,, ,[62,,,. . . .
/
." ,
/'::'~*- '~"4
\
,
{;;
"\'-~-- " ~ ' f - I - " , ' - 4 N J .... ~T~-, I
1'/
/'ki,._l_
;"
t
t 31501] -f 1116151~
/ .........
I
315e~;q
..sE523J/
3 [6513----~----~-e-" . - - - - - - - - - - - - - - - - - - - - - - - - ] -°-t~"/ I I [ 5 0 5 - - : : i ~ / - ' : - e - e - - - - - 4 ~ - - - - -e-~---- -e-o- --- -,~4~ ----~,-oI [ 660]'/~
I [660]~ 111502] I
166yBlTOyB 172yBI74yB I7~6yB178HFf8°H F
Fig. 16. U n p e r t u r b e d s i n g l e - n e u t r o n levels obtained by ref. 46). T h e asymptotic q u a n t u m n u m b e r s (2~2)[Nn=A]are indicated. T h e levels s h o w n are for t h e o d d - N n u c l e u s with one less n e u t r o n t h a n t h e nucleus indicated. Circles indicate levels filled in t h e limit o f zero residual interaction. Vertical bars are twice t h e odd-even m a s s difference.
K=O
EXCITED
BANDS
-YB AHF
LI.I 1.0 .6 z r'n .2
,B
,O
•O
2 +
ACE)
,B,D "G IB
B,(C) • (D)
,[E)
0 Io
.6
.2
4 +
-CO) • (D)
,B
,B "0
r
i
.I I
i
/
98 I00 102 104 106 RESIDUAL NEUTRON NUMBER
96
Fig. 17. Ratios o f t h e s u m m e d cross sections for J = 2 a n d 4 states in excited K -- 0 b a n d s to t h e s u m m e d cross section for t h e b a n d h e a d . Letters to the right o f the points indicate b a n d m e m b e r s h i p (see table 4). States w h i c h are doublets or are n o t resolved f r o m k n o w n 2+2 a n d 3 - states are n o t shown.
(p, t) REACTIONS
241
where X is the strength of the interaction, r~ is the vector to particle i a n d f ( r 1 , r2) is some function of the radial coordinates. These collective excitations are expected to appear in the energy gap. In general they are not expected to have more than a few percent of the ground state strength in two-neutron transfer reactions 14). (ii) Spin-quadrupole excitations a 9) are due to interactions of the type
--tcf(rl, r2)(~1 Y2(1))~(tr2 Y2(2))2,, where a~ is a Pauli spin matrix. Such an interaction can also result in 0 + states within the energy gap. It is expected 40) that the spin-quadrupole force will reduce the (p, t) strength to states of type (i). (iii) Pairing vibrations are due to the residual pairing interaction and should appear above the energy gap. The strength of the two-nucleon transfer reaction to such excited states is normally expected to be only a few percent of the ground state strength in the rare earth region 41). (iv) A combination of the monopole and quadrupole pairing interactions acting on particles moving in Nilsson single-particle levels with a nonuniform distribution of prolate and oblate orbitals around the Fermi surface has been shown 14, 42) to give excited 0 + states with (p, t) cross sections on the order of 10 to 20 ~ of the ground state strength in the actinide region. The cross sections are stable against changes in the neutron number. (v) In regions of changing equilibrium deformation the (p, t) reaction may populate a deformed 0 + excited state strongly in a nucleus with a spherical ground state if the target nucleus is deformed. Such states have been seen in Nd [ref. 43)], Sm [ref. 1)] and Gd [refs. 4, 6)] near N (neutron number) = 88. The 12 ~ state in 166yb (N = 96) fits in well with the systematics observed by Maher et aL 5), Else et al. 6) and Fleming et al. 4). They observe Gd, Dy and Er isotopes over the range 90 _< N < 96 which have excited 0 + states populated by (p, t) with 10 to 20 ~ of the ground state strength. The fourth mechanism described above might explain such systematics 14). The very strong states in 174yb and 176Hf (both N = 104) do not follow such a systematic trend. Nor do the other mechanisms for strong 0 + states apply since the H f and Yb nuclei are well deformed and have an approximately uniform distribution of oblate and prolate orbitals. These states might be explained 2) in terms of the pairing phase transition described by B~s and Broglia 41), Broglia et al. 44) and B~s et al. 45). In these papers it is shown that if there is a gap in the single-particle spectrum near the Fermi surface on the order of twice the energy gap parameter A and if there is sufficient degeneracy above and below the gap, appreciable strength will appear in the L = 0 two-neutron transfer reaction. The gap in the single-particle spectrum may be said to cause a phase transition from superconducting to normal. In fig. 16 the single-neutron levels determined by Ogle et al. 46) from the experimental energies of one-quasiparticle states are shown for several Yb and H f isotopes. The vertical bars indicate values of 2A. The condition for strong excited 0 + states is
242
M.A. OOTHOUDT AND N. M. ItINTZ
best met for 176yb and 178Hf targets. Using the parameters shown for 176yb(p, t) and calculations for a simplified two-level model 45) with a pair degeneracy of five per level, we find as a crude estimate o-~x/O-g.~"~ 0.67, whereas experiment gives 0.26. Large excited state cross sections are also predicted for some of the other Yb nuclei. The discrepancy between this model and experiment is not surprising, since the two-level model is certainly an oversimplification of the nuclear structure. Reaction effects have also been ignored (see sect. 5). Thus while we have a plausible mechanism for the strong population of 0 + states by (p, t) that applies to the Yb and H f isotopes, we do not have a detailed quantitative explanation. In fig. 17 the ratios of cross sections of the 2 + and 4 + members of excited K -- 0 bands to their bandheads are shown. Several states not resolved from known members c f gamma and octupole bands are not shown. A comparison of figs. 14 and 17 shows that the ratios for the excited bands are a factor of 1.5 to 2 times the ratios for ground bands. This increase in ratios might be due to mixing with nearby v-vibrational bands. A number of previously reported 0 + states were either not observed in this experiment or were observed With angular distributions different from the normal L = 0 shape. In the former class is the 1199 keV 0 + state 30) in 178Hf. A state is observed at 1272 keV near the 2 + member of this band, but in view of the low population of the bandhead (upper limit of 0.5 ~ of the ground state strength) the 1272 state is more likely to: be the 3 + member of the ;~-vibrational band. The 1434 keV 0 + state 3 o) is not seen either, although it is possible that this state was not resolved from the 1444 keV 0 + and their sum is the 1448 keV state reported here. However, the peak shape in the (p, t) spectrum does not indicate a doublet. In 174yb a 0 + state was reported 31) at 1305 keV in 7-ray work using a NaI detector. The (p, t) spectrum was carefully searched and an upper limit of 0 . 4 ~ of the ground state strength was placed on this state. If, as suggested by Wilson and Pool 31), the 1228 keV 7-ray depopulating this state corresponds to the 1245 keV 7-ray reported by other workers 47), the state reported as a 1305 keV 0 + is actually the 1316 keV 2-2. The 1410 and 1628 keV peaks observed in 17Zyb are near the 0 + and 4 + members of a K = 0 band at 1405 and 1632 keV reported in (n, ~) work 32, 4s). (The 2+0 at 1476 keV would not have been resolved from the strong 7-vibrational bandhead.) T h e angular distribution for the 1410 state does not have the oscillatory L = 0 character, although it is possible that for such a weak state (less than 0.5 ~ of the ground state strength) the minima could be filled in by the normally insignificant triton background. Peaks at 1225 and 1568 keV in ! 7Oyb are near 0 + states reported at 1228 and 1566 keV in 7-decay work 33). Again these states do not have an L = 0 character. The 1568 keV peak is quite strong (3 ~o of the ground state strength) so that an L = 0 character should have been observed. A peak at 1634 keV near the 2 + member of the 1568 keV band was also observed with about 30 ~ of the 1568 keV state strength. Without
(p, t) R E A C T I O N S
243
convincing evidence for a 0 + at 1568 keV, we cannot assign this peak as a 2 + on the basis of the shape of its angular distribution. I n . l e a y b peaks at 1161 and 1231 keV are near 0+0 (1156 keV) and 2+0 (1234 keY) states reported in ?-ray and conversion electron work a4). The 1161 keV peak does not have an L = 0 character, although for such a weak state background might
GAMMA VIBRATIONAl BANDS 16
"',
", 14
GROUND STATE BANDS /
G A M M A ~ "- - " ~ -
BANDS
"Fv
~
12 i
>-
1.5
l
i
1
i
i
.YB •
"-.
o Ld
~od '-? bJ O2
4
m~
2 •
f
i
i
i
I
i
r
i
~
~
t
60
U3
20 ~0
.~].~ b~
6
2 , 9c8 I00 , , ' 96 102 104 I'06 RESIDUAL NEUTRON NUMBER
Fig. 18. Properties of 7-vibrational bands. Inertial parameters, excitation energies, B(E2) values 37.65) and summed (p, t) cross section are shown.
fill in the minima. The 1231 keV peak is twice as strong as the 1161 keV peak (unlike other 2+0 states which are at most equal in strength to their bandheads) and does not have an L = 2 shape. 4.2.3. ?-vibrational bands. Bands with positive parity and K = 2 may be interpreted as quadrupole vibrations of the nuclear surface, or microscopically as superpositions of two-quasiparticle states 49). States were observed at the energies of previously reported ?-vibration bandheads with 5 to 10 % of the ground state strength. Angular distributions for these states (see fig. 3, 1636 keV in l~4Yb and 984 keV in 168yb ) are more oscillatory than those for the 2 + members of the ground state band. In spite of the general instability of shapes for angular distributions, we have as-
244
M.A. OOTHOUDT AND N. M. HINTZ
signed these states to be the 2 + bandheads for y-vibrations on the basis of their excitation energies. A candidate for the previously unobserved y-vibration bandhead in 166yb has been found at 931 keV. The 4 + and, in some cases, the 3 + members o f these bands were identified by comparison with the known excitation energies of these states. The 4+2 of ~76Hf is assigned to the state at 1545 keV using the ground state band moment of inertia and the I(I+ 1) energy spacing rule. (The use of the ground state band moment of inertia is justified in fig. 18.) States of unnatural parity such as 3 + are not expected to be excited except very weakly by the (p, t) reaction (see subsect. 4.1). The excitation mechanism may be a two-step process such as (p, t) to the 2 + bandhead followed by inelastic excitation to the 3 + by the exiting triton or perhaps inelastic excitation of the target by the incoming proton to the 2 + of the ground state band followed by an L = 2 (p, t) transfer to the 3 +. In fig. 18 cross sections and other properties of these bands are shown. The cross sections for the ~7°yb 2+2 and 4+2 are upper limits only, since they are not resolved from the 2+0 and 4+0 members of the 1072 keV K = 0 band. For the H f and heavier Yb nuclei, B(E2) values are low and excitation energies high, indicating a low degree of collectivity for these states. The (p, t) cross sections do not follow these trends. In addition to these K = 2 bands, a number of other possible 2 + states were identified by the shapes of their angular distributions with up to three-quarters of the yvibration bandhead strength (see table 4). Since no 0 + states were observed below them with a reasonable energy spacing, they are assumed to be J~K = 2+2. These states could also be 2+1, but angular distributions for known 2+1 states do not resemble either y-vibrational or ground state 2 +. The 1604 keV state in ~V2Yb is assigned by Burke a7) as a K = 2, {~[512] -½1521]} two-quasiparticle state, but by the Z-A selection rule such a configuration should not be observed in (p, t). Thus the strength of the 1604 state is either a measure of the mixing of other configurations into this state (e.g., ½[521] + ~ [521]) or a measure of the breakdown of the Z-A rule due to the finite value of fl as discussed in subsect. 4.1. 4.2.4. Octupole vibrations. Several peaks were observed near the energies of previously identified octupole states. The shapes of angular distributions were not distinctive enough to assign L-values. Assignments are therefore made only on the basis of excitation energy. Theoretical calculations by Neerg~rd and Vogel 5 o) show that these are not very collective octupole states, since most of the E3 strength is expected to be at very high energy; only a small fraction is brought down below 2 MeV by the spin-orbit part of the nuclear potential. They also predict that for Yb and H f nuclei, the lowest octupole states will be K = 1 or 2. Due to strong Coriolis mixing, however, the K quantum number is not expected to be very good s o). In fig. 19 excitation energies and summed cross sections are shown for 3- states observed. The K-assignments are shown where known from other experiments. The 3 - states are excited with 2 to 5 ~ of the ground state strength. In 172yb members of a K = 1 band are seen with J~K, excitation energy and sum-
(p, t) R E A C T I O N S
245
med cross section of 1-1, 1154 keV, 14 #b/sr; 3-1, 1218 keV, 18 #b/sr; and 5-1, 1350 keV, 9 #b/sr. It is also possible that the 1536 and 1556 keV states are 6- and 7members of this band 51). The 1556 keV state is, however, near the 3+2A at 1549 keV. The alternative assignment of O'Neil and Burke 51) of 7- to the 1536 keV state seems more likely, since a 6- with unnatural parity and high L-value would not be expected to be populated with much strength. In 174yb the 2- (1316 keV) and 3(1386 keV) members of a K = 2 band are seen with summed cross sections of 3 and 38 #b/sr respectively. Burke and Elbek 37) find an inertial parameter h2/2J = 10.0 keV for this band and predict the 5-2 at 1564 keV near the 2+0B. Excitation energies
3" S T A T E S 2.0 A
AHF e2
• YB
>
•
*0 v
>..
1.5 •
Or" L.d
.2 A2
A2
ol
Z laJ
1.0 I
f
I
•
•
!
I
I
I
I
62 ~,2 •
A2
•2
I
I
"~ so "~
zL IE
40
20
b~
,o
io 6 4 I
I
96 98 I00 102 RESIDUAL NEUTRON
104 106 NUMBER
Fig. 19. S u m m e d cross sections a n d excitation energies for 3 - states. N u m b e r s to the right o f the p o i n t s give K q u a n t u m n u m b e r s . T h e 16Syb points are u p p e r limits only, since they are n o t resolved f r o m the 4+0B a n d 2 + 0 D states.
from this experiment give h z / 2 J = 11.7 keV and the 5-2 at about 1596 keV. Near this energy (1608 keV) we find a weak peak (summed cross section of 4.6 #b/sr) which may be the 5-2. However, due to the strong Coriolis mixing predicted 5 o) for octupole bands, it is not clear whether a simple I(I+ 1) rule is sufficient to predict excitation energies. Several other 1- states are indicated in table 4. 4.3. O D D - N Y b T A R G E T S
The study of the odd-N 173yb and 171yb targets with the (p, t) reaction was more difficult than the study of the even-N targets. Experimentally, the high density of weak states populated at low excitation energy in the odd-N nuclei (see figs. 20 and 21) made resolution of peaks difficult. Interpretation of the spectra was als0 more diffi-
246
M . A . OOTHOUDT AND N. M. HINTZ
cult, since the total angular m o m e n t u m of the final state is not uniquely determined by the L-value of the transfer when the target angular m o m e n t u m is not zero. For ~
,
,
~
~
J
J
i,
i
i
i
~
,
I
'
d
;
E
j
,
,
J
173yB(p,t) 17tyB tE[5os] !. I
]
x_
I000
~"
I
i
5 ~BAND
,
~
=
,
i
J
[
11521]
'
'
7 [6s31 i
5 [512]
5[~i~ I
: i :J
(I-) BAND 5[72SI]
I00
~
, ; ]i
1,4 I0
,
~
r
J
:
J
2.5
, I
L
2.0
~
i
I
t
1.5 ( MeV )
Ex
]1
,
i
,
1.0
I 0.5
0
Fig. 20. Schematic spectrum for 17Syb(p, t)171yb. For each state the summed cross section is plotted at the proper excitation energy. The sum is over standard angles. The major single-particle components o f rotational bands are indicated by giving the asymptotic quantum numbers (2#2) [Nm_A]. The " 5 - band" and " ( 1 - ) band" are possible bands built on vibrations of the even-N core. i
i
i
171YB ( p , t ) 169yB 7 E6ss] 5 [5~2] I
I000
I
I
L
-i
o~ t'b
I [510]
IO0--
s[521]
4 ! t
b %1
tO:L_ t-
i
2.5
i
J
2.0
i
i
t
t
I
L
1.5
~
,
~
i
1.0
I
,11 i
i
0.5
Ex ( M e V )
Fig. 21. Schematic spectrum for t71yb(p, t)169yb. See caption to fig. 20. Possible ½-- states populated by L = 0 transfers are indicated by (1).
this reason only dominant L-values assigned with C U R V E F I T are given for odd-N nuclei in this experiment. In general, we assume that the (p, t) cross section decreases rapidly as L increases as for the even-N nuclei. Thus we assume if a member of a
(p, t) REACTIONS
247
band allowed by selection rules is not seen, higher band members will be too weak to be seen. However, the operation of the A selection rule can modify this assumption. A further problem for the 173yb target is that with J~ = ~ - it is usually possible for the reaction to proceed by more than one L-value. (For 171yb(p ' t) with ground state J~ = ½-, parity and angular m o m e n t u m conservation allow only one L-value.) For example, the strong " L = 0 " transfer to the ~[512] state at 122 keV in 171yb may go by L = 0, 2 or 4. An extensive angular distribution was taken on the 122 keV state to test the contribution of L-values other than 0. An option in C U R V E F I T was used which adjusts the sum of two standard curves to the data to give the best Z 2 per point. The best fit (Z z = 4) was obtained from the sum of the 0 + and 2 + members of the 1 7 4 y b ground band. The i 7 4 y b ground state L = 0 by itself gave Z2 = 112. Adding together the 0 + and 4 + members of the ~74yb ground state band did not give as great an improvement (X z = 30) as the 0 + and 2 + combination. To test whether this improvement was fortuitous, the 16syb ground state (the L = 0 most similar to the 122 keV state) was also fit with the same combination of states, TABLE 5a States in 171yb This experiment Ex (keV)
0 75 94 123 168 208 228 248 318 369 448 486 502 601 778 831 907 940 960 995 1027 1048 1085 1111 1140 1198
Other experiments
~tr . . . . ") (#b/sr)
L
5 6.8 0.7 5 ¢) 27.7 1.6 7 9.4 1.0 5 484.6 5.5 5 6.9 0.6 5 90.2 2.1 5 c) 6.7 1.1 5d) 18.6 1.6 5~) 58.1 2.1 7 1.7 0.3 5 13.0 1.0 5a) 2.8 0.8 7 2.3 1.1 5 a) 7.0 0.6 7d) 1.7 0.5 7 2.2 0.4 5 a,~) 14.7 1.6 5 6,0 0.7 5 29.3 1.7 7 17.9 1.2 5 17.0 1.1 5 7.1 0.8 7 14.4 0.8 7 10.6 0.8 5 ~,d) 6.4 0.7 7 3.9 0.5
2
0 2 (4) 2 (4) (4)
(4)
E,, J'rrK" Ref. (keV) b)
0 1--1 75 5--1 95 7+7 122 5--5 168 9+7 208 7--5 230 7--1 247 9--1 317 9--5 369 13+7 450 11--5 487 1 1 - 1 509 13-1 604 13-5 779 15-5 832 15-1 902 3-3 945 1-1 995
3-1
•052 5-1 1079 7-3 1114 13-11 1144 7-1 1189 19-5
57) 27) s7) 27) ST) sT) sT) SS) sT) 2s) 2s) 2S) SS) 2S) S8) 5S) 6O) 60)
Other experiments
This experiment
Ex (keV)
1 2 4 2 7a) 1265 5 a) 1338 7d) 1389 5 1436 7 1487 7 1513 5e) 1 5 9 0 5e) 1624 5 1649 7 1673 7d) 1698 7 1736 7 1763 7 1827 7 1845 7 1868 7 1902 10 1966 10 60) 1995 10a) 2108 10 60) 2286 10 60) 2303 10 SS) 2373 10 60) 2476 10 2S) 2642 10
~,Ge.m" a) ~b/sr) 6.3 2.9 2.7 46.9 8.0 7.3 65.2 17.7 9.9 8.2 1.9 9.1 5.0 7.3 5.8 6.2 4.2 4.1 6.7 18.6 13.8 8.0 5.2 11.4 13.9 6.7
0.7 0.6 0.6 1.9 0.8 0.9 1.9 1.0 0.9 1.0 1.3 1.0 0.9 0.8 1.0 0.9 1.3 1.6 1.0 1.2 1.3 1.4 1.0 1.2 1.1 1.4
L
Ex J'rrK" Ref. (keV) b)
1266 15-11
ss)
1436 17-11
ss)
1626 19-11
5s)
1834
5s)
(2)
0
(0)
21-11
248
M . A . OOTHOUDT AND N. M. HINTZ TABLE 5b States in 169yb This experiment Ex
N a . . . . a)
(keV)
(/zb/sr)
27 5 71 5 87 5 99 5 194 7 245 5 268 5d) 484 10 516 5 570 5 659 5 750 5 808 5 1033 5 1108 5 1169 5 1350 5 e)
349.0 6.4 43.1 130.0 1.6 11.5 27.6 1.7 3.5 10.4 33.4 5.5 17.3 9.7 6.1 4.9 27.8
4.6 1.5 5.7 8.4 0.3 1.6 2.9 0.2 0.4 0.8 1.0 0.6 0.7 0.5 0.4 0.5 1.1
Other experiments L
0 2 2 (4) 4
2
Ref.
Ex
J'~rK"
(keV)
b)
24 71 87 99 191 244 264 487 512 570 660 747 807
1-1 9+7 3-1 5-1 5-5 7-1 9-1 11-1 13-1 5-5 3-3 9-5 15-1
59) 59) sg) 59) sg) 59) s9) s9) 59) 61) 61) 61) sg)
1111
5-5
61)
1353
3-1
6z)
This experiment Ex
Z a . . . . a)
(keY)
(#b/sr)
1426 5 1463 10 1479 5 1513 10 1528 5 1582 5 1621 5 1714 5 1740 5 d'¢) 1785 5~) 1943 7 1971 7 1997 7 d) 2053 12 2129 12 2286 12
10.8 2.0 4.3 5.3 9.8 20.2 9.3 7.4 14.0 13.5 10.7 9.7 4.7 10.4 9.0 6.3
0.7 1.4 0.7 1.1 0.8 1.5 0.9 1.2 0.9 1.1 1.3 1.4 1.3 0.8 1.1 1.5
Other experiments L
E~
J'~rK" Ref.
(keY)
b)
1466
7-1
61)
1616
3-
64)
0
(0) (o)
") Sum of center of mass cross section over standard angles. b) j , = 23, K" = 2 K .
°) Possible doublet. d) Contaminant subtracted. e) Doublet.
giving X2 = 57 for the 174yb ground state by itself and 48 for the sum of the 0 + and 2 + states. Since the improvement in Zz is so much more dramatic for the 122 keV state than for the ~6Syb pure L = 0 state, we conclude that the 122 keV state does have a significant amount (15-20 ~ ) o f L = 2 in it. In general the strongest L = 0 transfer for odd-N nuclei goes to a one-quasiparticle state with the same quantum numbers as the target ground state. The (p, t) amplitude is then a coherent sum over zero-coupled pairs o f neutrons picked up from the even-N core. The strong L = 0 states are at 122 keV in 171yb and 24 keV in 1 6 9 y b . The strength of these states is respectively 54 ~o (including any L = 2 and 4) and 37 (pure L = 0) of the average strength of the ground state transitions in neighboring even-N nuclei. A similar loss of strength in the odd-N Sn isotopes has been attributed to blocking by the unpaired neutron s 5). The (p, t) reaction may be expected to populate one- and three-quasiparticle states and states formed by coupling a quasiparticle (the unpaired neutron in the target ground state) to vibrations of the even-N core. Solov'ev et al. 56) have calculated the mixing of one-quasiparticle states with quadrupole and octupole vibrations of the even-N core. We will use their descriptions for the states throughout this section. The notation for these bands is f2[NnA] ® Q(2, #) where the first factor is the
(p, t) REACTIONS
249
asymptotic quantum numbers of the single-particle state and Q(2, #) is the vibration with multipolarity 2 and projection on the symmetry axis #. The bands thus formed have K = O__/~. The data are presented in table 5 and figs. 20 and 21. In the table excitation energies and errors (keV) are given in the first two columns. Columns 3 and 4 give the sum of center of mass cross section over standard angles and its relative error. Column 5 gives dominant L-values assigned in this experiment using CURVEFIT. Columns 6 and 7 give excitation energies (keV) and assignments from other experiments. The quantum numbers for the assignments in column 7 are J ' = 2J and K' = 2K. "Contaminant subtracted" means a state from another Yb isotope has been subtracted to give the cross section indicated. In the following paragraphs we discuss the bands observed. 4.3.1. The ½1521] band. In 171yb the bandhead (which is also the ground state) has a summed cross section of 7 #b/sr even though the I - A selection rule inhibits population of the state (subsect. 4.1). A possible mechanism for exciting this state is to pick up a neutron pair coupled to L = 2, leaving the even-N core in a v-vibrational state coupled to the unpaired neutron: 31512] ® Q(2, 2). Calculations s6) indicate that such a configuration contributes less than 1 ~ to the band. Members of the band up to perhaps the ~5 state were identified by comparison with known excitation energies 57, 5s). The weak _1~ member of the band would not be resolved from the ~5_member of the ~ [633] band. In 169yb the lowest strong L = 0 transition goes to the bandhead of this band at 24 keV. Band members up to perhaps ~5_ were identified by comparison with (e, 2n) data 59). The peak identified as the -~ member may contain strength from the member of the ~ [512] band, while the .~5_peak may contain significant strength from the ~ member of the 31521] band. 4.3.2. The -~[633] band. The ~, 9 and ~ - members of this band are seen in 17zyb. The ~ - and ~ - members would not be resolved from other states. In 169yb this band is the ground state band. The bandhead is not observed in agreement with the I - A selection rule. The ~ member of the band at 71 keV is populated but is very weak and is not well resolved from the much stronger states at 87 and 99 keV. A state near the expected 59) position of the _~3_member of the band at 270 keV is probably the 9 member of the ½1521] band since it is unlikely that the .½a_member of the band (which is populated by L = 7) would be observed when the @ member (L = 5) is not observed. 4.3.3. The 31512] band. This is the ground state band in 173yb and therefore gives the strong L = 0 transfer in the (p, t) reaction to 171yb as discussed above. Band members are seen up to the ~ state. The ~ member of the band appears to be a doublet in the (p, t) spectrum, although there are no other known states in this energy range. In ~69yb the bandhead is observed even though its population is inhibited by the I - A selection rules. The ~ band member would not be resolved from the 29-member
250
M. A. OOTHOUDT AND N. M. HINTZ
of the ½1521] band. A peak is observed near the location of the ~ - member of the band but since lower members of the band are not seen, it is probably due to the i a member of the ½1521] band. 4.3.4. The ~[521] band. Peaks at 907 and 1085 keV in 171yb are near the previously reported 6 o) ½ and ~ members of the a [521] band. The identification of these peaks is in doubt, however, since the s and ~ members of the ~ [523 ] are near these energies. Furthermore, the ~ member of the -~[521] band (which should appear near 981 keV assuming an I(I+ 1) spacing for the band) does not appear. In 169yb candidates for the s and ~ members of the band are observed at 659 and 808 keV. Calculations 56) show this band has a 4 ~o ½1521] ® Q(2, 2) component which might be expected to be populated in (p, t) through excitation of the v-vibration of the core. The s state 61) at 721 keV is not observed. The peak identified as the state may contain some strength from the ~ member of the 5 [523 ] band. The ~ member of the ~ [521 ] band may contain strength from the _1~_member of the ½[521] band. The bandhead of the lower ½1510] band is also near the ~ member of the ~[521] band 62), but the transition to the ½ state is 27-A hindered and no higher members of the ½1510] band are seen. Thus the ½ state would not be expected to contribute Strongly to the peak. 4.3.5. The ½1510] band. In a71yb population of this band requires pickup of a particle from a level well above the Fermi surface. Occupation of such levels may be expected to be low, so cross sections might be expected to be small. However, calculations 56) show that this band is 51 }o -~[512] ® Q(2, 2). This component might be populated strongly by picking up a pair coupled to L = 2 from the target, leaving the core in a v-vibration that is coupled to the unpaired s[512] neutron. Members of the band up to ~ are observed; the ~ member may contain strength from the -t2-smember of the ~-[505] band. The sum of band members seen is 37/~b/sr, which is an upper limit for the total L = 2 strength for the band. The strength of the v-bandhead in the neighbouring even-N nuclei is about double this. This loss of strength is expected from the splitting of the strength between the ½ and -~ states formed by coupling the vibration to the -~-. Further redistribution of strength may occur due to mixing of the resultant ½- state with the ½1510]. A candidate at 1389 keV for the second ½- state resulting from this mixing is discussed in subsect. 4.3.8. Members of the ~- band were not identified. In a69yb bandheads for fragments of the ½1510] state have been reported 62) at 813 and 1317 keV. At 808 keV in the (p, t) spectrum, a peak is seen with summed cross section of 17 #b/sr. Since the 2J-A selection rule inhibits population of the ½state and no higher band members are seen, this peak is probably the ~ member of the 31521 ] band and/or the -~- member of the ½1521 ] band. Peaks are observed at 1350 and 1463 keV near the energies of the -~ and ~ members of the 1317 keV band. The ½ member of the ½1510] would be inhibited by the Z-A selection rules, but the unobserved s member would be allowed. 4.3.6. The~}[523]band. In ~71Yb the 2~-A selection rule inhibits population of
(p, t) R E A C T I O N S
25i
the bandhead. A 15 #b/sr peak is seen near the expected energy of the bandhead (907 keV) but it may be the bandhead of the 31521] band. Peaks at 995, 1085 and 1I98 keV are reasonably dose to the previously reported 6a) ~, 29_and - ~ members of the 3[523] band at ~ 987, 1083 and 1208 keV, although other states couldcontribute to these peaks. The 3 and 9 members of the ~ [523 ] band are observed in ~69yb at 570 and 750 keV. The ½ member has been previously reported 64) at 648 keV; the nearby bandhead ofihe 31521] band may obscure this state. 4.3.7. The -111505] band. Data from the (a, 3n) reaction s8) indicate the presence 11 15 of an J~-[505] band in 171yb with members at 981 (-~-), 1114 (~-), 1266 (-~), 1436 (-~-), 1626 (~-) and 1834 (21_)keV. In the (p, t) spectrum peaks are seen very close to all these energies except for the ~-~ bandhead. This band has not been identified in 169yb" 4.3.8. States based on collective vibrations o f the even-N core in 171yb" An L -- 0 transfer a t 1513 keV ~ndicates a ~- state with 13 ~ of the summed strength of the 3- at 1:22 keV. To find higher band members we use an inertial parameter of hZ/2J 11.4 keV similar to that of the ~[512] band. We find candidates for a at 1590, a 9 at 1698, an ~1 at 1827 and a _13_at 1966 keV. It is possible, however, that the 1698 and 1827 keV peaks belong to other bands. Since the ~- Nilsson orbitals near the Fermi surface have already been accounted for, this band probably is a K = 0 Vibration of the core coupled to the ~[512] unpaired neutron. Surprisingly, the 1513 keV state is considerably stronger than the excited K = 0 bandheads in adjacent even-N nuclei. A strong, possible L = 2 transfer to a peak at 1389 keV indicates a state with total angular momentum of k, 3 , . . . , or 9. A low value for the angular momentum of the state is favored by the fact that the ~72yb(d, t) reaction 60) populates the state whereas the t72yb(3He, a) reaction 63) does not. It is interesting to speculate that this state may be the other K = ½ band resulting from mixing the ½1510] with the ~[512] ® Q(2, 2), as discussed in subsect. 4.3.5. The 1436 and 1487 keV states could be the 3 and 3 members of the band although they give inertial and decoupling parameters somewhat higher than average for vibrational states in this region 66). Peaks seen at'1590, 1673 (or 1698) and 1868 keV could be higher band members. 4.3.9. Other states in 169yb. Three weak transitions that may be L = 0 are observed at 1513, 1582 and 1740 keV. Sincethe target is ½-, these states are necessarily ½-. Assuming inertial and decoupling parameters similar to other nearby ½bands, states at 1621 and 1785 keV could be the 3 members of the 1582 and 1740 keV bands respectively. 4.4. N A T U R A L T A R G E T S
Even-Z nuclei from Gd through Os were investigated to find 0+ states strongly populated by the (p, t) reaction. Targets consisting of elements with the natural abundances of isotopes were exposed to the i9 MeV proton beam and tritons were
252
M . A . OOTFIOUDT AND N. M. HINTZ
collected with nuclear emulsion plates. D a t a were t a k e n only at 27½ ° near the L = 0 m a x i m u m . The lowest L = 0 transfer for each nucleus was identified using twon e u t r o n s e p a r a t i o n energies ( $ 2 , ) f r o m mass m e a s u r e m e n t s 29, 67-71) a n d intensity systematics. Excited states were identified using k n o w n level schemes. F o r e v e n - N nuclei only m e m b e r s o f the g r o u n d state b a n d u p to 4 + o r 6 +, 7-vibration b a n d h e a d s , 3 - o c t u p o l e states a n d excited 0 + states were identified with a n y certainty due to the
(p,t) GD
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NATURAL ER
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RESIDUAL NEUTRON NUMBER Fig. 22. Cross sections for states populated by the (p, t) reaction at 27½° and Ep = 19 MeV using natural targets (except for Yb and Hf). Cross sections are upper limits only. The upper figure shows ground bands, while the lower shows excited bands. For excited 0 + states, the sum of all known 0 + states observed is plotted. Relative errors are shown only where they are larger than the data points. Points plotted are 0(0+), A(2+), 11(4+ or 3-) and ×(6+). h i g h density o f states p o p u l a t e d . F o r o d d - N nuclei, only the lowest L = 0 transfer was identified; it was assumed t h a t the lowest L = 0 transfer goes to the one-quasiparticle state with the same q u a n t u m n u m b e r s as the target g r o u n d state. The quasiparticle assignments o f B u n k e r a n d Reich 66) are used here unless otherwise noted. Cross sections for the states o b s e r v e d are p l o t t e d in fig. 22. F o r completeness, d a t a t a k e n with isotopic Y b a n d H f targets are also shown, In general the cross sections are u p p e r limits only, since the high density o f states could p r o d u c e p e a k s
(p, t) REACTIONS
253
composed of states from different nuclei. Relative cross sections between isotopes of the same element were obtained using the natural abundances of the isotopes 17). Normalization between elements (except for Os) was obtained using cross sections from recent (p, t) experiments 4, 5, 7) at 17.5 and 18 MeV. No corrections were made for the differences in proton energy, since DWBA calculations similar to those in sect. 5 predict a change in the cross section for 0 + states near the L = 0 maximum of less than _+ 10 ~ over a range of proton energies from 17 to 20 MeV. For Os, elastic proton scattering was done at 50 °, 60 ° and 70 ° and the cross sections were assumed to have the same ratio to Rutherford as 176yb(p, p). It is not expected that the accuracy of the normalization between elements is better than 25 ~ . The cross sections obtained for the lowest L = 0 transfers are given in table 2 along with Qvalues measured. Since the thickness of the targets and carbon backings were not carefully measured, absolute Q-values could not be obtained, although relative Q-values should in general be good to _ 5 keV. The measured Q-values for all isotopes of an element were shifted to agree with accurately known Q-values 29, 68, 69). The relative cross sections measured agree reasonably well with recent data taken with isotopic targets 4- 7, 72). The previously uninvestigated Os isotopes do not seem to have any excited 0 + states with greater than 10 ~ of the ground state strength up to an excitation energy of 2.5 MeV. In the paragraphs below, contaminants in the targets are discussed before the rare earth elements of interest. Cd The natural Os target contained a considerable amount of natural Cd. The presence of the Cd peaks allowed an independent check of the Q-value determinations. The Q-values were shifted for this target to give best agreement with Szn values 6s) for 192Os and 19°Os. The resulting San values for the Cd isotopes (table 2) are in good agreement with previously determined values 71). Ta The Gd, Dy and Er targets contained a significant amount of Ta from either the Ta powder used to reduce the oxides or from the Ta evaporation boat. A plate was exposed with a pure Ta target at 27½° to identify possible contaminant states. The major Ta states seen up to 2.5 MeV and their strengths relative to the ground state are 0 keY (1.00), 137 keV (0.06), 296 keV (0.03), 1520 keV (0.06), 1549 keV (0.01), 1732 keV (0.02), 1892 keV (0.01), 1975 keV (0.02), 2134 keV (0.01) and 2337 keV (0.02). The 0, 137 and 296 keV states have been previously identified 73) as members of the K ~ = 7+ ground state band. The 6 ~ 1520 keV state might be a K = 0 core vibration similar to the 4 ~ state excited by (p, t) at 1776 keV in the even-Z isotone 178Hf. By assuming that ratio to Rutherford for Ta is the same as for 176yb(p ' p) and doing elastic scattering at 27½°, an absolute cross section of 354 #b/sr (_+25 ~ ) was obtained. This is somewhat lower than the adjacent even-Z nuclei. If this value is correct, it indicates the odd proton has a surprisingly large effect on the neutron pairing. Such a reduction has been seen 23) for 1°9Ag(p, t) relative to 10Spd (p, t). Au One of the W targets was made on an Au backing. In a plate exposure with
254
M.A. OOTItOUDT AND N. M. HINTZ
a pure Au target, the major states observed below 2 MeV and their relative strengths are 0 keV (1.00), 261 keV (0.02), 550 keV (0.03), 828 keV (0.02), 1349 keV (0.01) and 1599 keV (0.02). The 0, 261 and 550 keV states have been previously identified 74) as -}+, ~+, and ~+ or 5+ states. The 1599 keV state might possibly be a K = 0 core vibration similar to the 4 ~o state at 1551 keV observed 7Z) in the (p, t) reaction to the isotone a94pt. An absolute cross section of 454 #b/sr (_+25 ~ ) was determined by the procedure used for the Ta target. G d - The absolute cross sections for the Gd isotopes were obtained by comparison with the 16°Gd(p, t) ground state cross section measured by Fleming e t al. 4) at 18 MeV. The Q-values were adjusted to give best agreement with S2n values 69) for 158Gd and 16°Gd. The lowest L = 0 transfer in 15aGd to the 31521] ground state 75) was not resolved from the strong 0 + state at 1050 keV in 156Gd. The 0 + state was subtracted from the peak to give the 153Gd cross section using the strength relative to the ground state given in ref. 4). No strong excited states other than those identified in refs. 4, 6) were observed. D y - Absolute cross sections were obtained by comparison with the cross section given by Maher e t al. 5) for the ground state transition in 164Dy(p, t) at 17.5 MeV. The Q-values were shifted to give agreement with S2n values 69) for 164Dy. The S2n value for 16~Dy disagrees strongly (18 keV) with the previous value 69). Since there are no other peaks in the proper energy range with sufficient strength for the L = 0 transfer, either the previous S2n value is incorrect or the excitation energy (178 keV) of the -}[642] state 66) is wrong. The ground state of the Ta contaminant obscured the 4+ member of the 16ZDy ground state band. Three relatively strong peaks were observed that do not correspond to states in the even-N isotopes observed in ref. 5). They could be states in 16~Dy at 1090, 1128 and 1270 keV with approximately 20 of the lowest L = 0 strength or states in ~S9Dy at 529, 567 and 709 keV with approximately 25 ~o of the lowest L = 0 strength. However, these peaks could equally well be sums of unresolved weak states. E r - The cross sections were normalized using the ground state transition cross section for 166Er(p, t) measured by Maher e t al. 5) at 17.5 MeV. The Q-values were shifted to give best agreement with the ~68Er ground state and the ~66Er first excited state San values 69). The 166Er ground state was obscured by the Ta ground state. A cross section for the ~66Er ground state was obtained from the measured ratio of the ground and 137 keV states of Ta. The a62Er ground state was obscured by the 1520 keV Ta state. The Q-value for 164Er(p ' t) was obtained from the first excited state and the cross section obtained by using the known cross section ratio of the 137 and 1520 keV states. The'lowest strong L = 0 transfer for 167Er(p, t) should go to the previously unobserved ~ [633 ] state. However, due to strong Coriolis mixing of the positive parity bands from the i~ spherical orbit, the 5 + state at 63 keY, which is mainly a rotation, based on the ~[642] band 76), may be expected to have appreciable (p, t) strength. In the (p, t) spectrum, a peak with a 96 #b/sr cross section appears near the expected location of the 63 keV state. A strong (293 #b/sr) peak at
(p, t) REACTIONS
255
462 keV is not identified with any known Er states and may therefore contain a major part of the ~[633] bandhead. In fig. 22 the sum of these two states is plotted. Preliminary (p, t) data 77) using an isotopic 167Er target Confirm that the 63 and 462 keV peaks are both states in ' 65Er populated by L = 0 transfers. Three strong peaks have not been identified with known Er states. They could be either states in 16SEt at 3016 keV (with 11 ~o of the ground state strength), 3057 keV (6 ~ ) and 3100 keV (20 ~ ) or states in 165Er at 1373 keV (with 18 ~ of the strength of the 462 keV state), 1414 keV (10 ~ ) and 1457 keV (32 ~). H f - Absolute cross sections were obtained from the 178Hf(p, t) data of this experiment. The Q-values were shifted to give best agreement with S2n values 29, 67) for the ground state transitions for 1s OHf and 178Hf(p ' t) and the transition to the 321 keV state in 177Hf. The S2n values for the 177, 176Hf(p ' t) reactions have been previously determined to an accuracy of only 42 and 59 keV respectively z9). The lowest L = 0 transfers for 177, 176Hf(p ' t) were found to lie within 20 keV of each other, but on the basis of cross section systematics and the location of the first excited states in the respective bands, it is possible to arrive at the Q-value assignments given in table 2. The (p, t) S2n value for 177Hf disagrees strongly (132 keV) with the previous value 29). A strong peak not identified with any known H f states could be a state at 1478 keV in a77Hf with 21 ~ of the lowest L = 0 strength or a state at 656 keV in 175Hf with 10 ~ of the lowest L = 0 strength. W - Absolute cross sections were obtained by comparison with the ground state transition cross section at 18 MeV for 186W(p, t) given by King et al. 7). A cross section determination similar to that done for the Ta target gave a normalization about two-thirds of the King value, in better agreement with the systematics for ground state cross sections (fig. 22). However, in view of the uncertainties in the proton-scattering method, we will use King's value here. The Q-values were shifted to give best agreement with S2n values 68) for 184W. The S2n values for 183W and 82W are not well known (see table 2) and identification of the L = 0 states for these nuclei was difficult. The final identification was made on the basis of other weak excited states in the residual nuclei. The values of $2. which we obtain do not agree well with previous values. The peak chosen as the L = 0 for 183W(p, t) is surprisingly weak, which may indicate an incorrect identification of the state or a real loss of strength due to blocking of the ½[510] orbital with its large (p, t) strength (see sect. 5). If the assignment is incorrect, the only possible alternative choice giving a reasonable strength to the aSaW(p, t) L = 0 transfer is to make it and the ~s2W(p, t ) l S ° w ground state transition an unresolved doublet. This choice, however, makes the 18 oW ground state quite weak. O s - Cross sections were normalized as desciibed above. The Q-values were shifted to give best agreement with $2. values 68) for 1920s and 19°Os. The value of $2, obtained here for 188Os does not agree well with previous values. The i87Os and 186Os(p, t) reactions were not identified. The peak identified as the v-vibration in ~86Os at 766 keV is unusually strong (19 ~ of the ground state strength) compared
256
M . A . O O T H O U D T A N D N. M. H I N T Z
to other y-vibrations. It is possible that this peak could contain strength from a state in another nucleus, but this would give as upper limits a 4 % state at 1734 keV in 19°Os, a 10 % state at 1334 keV in 188Os or a 25 % state in 187Os at 1138 keV. It is possible that strong Os states are obscured by the Cd contaminant peaks.
iO s
L
"TeYb(p,t) Ep =lgMeV CC
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102 ==~', ,,,,.
-~
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60
90
120
150
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The W and Os nuclei span the shape transition from deformed to spherical near N = 110. For the Sm nuclei the shape transition near N = 88 results in 0 + states excited by (p, t) with over 57 % of the ground state strength 1). In the W and Os nuclei, however, the strongest 0 + states observed have less than 10 % of the ground state strengths. The lack of strong excited 0 + states may be due 72) to the more gradual shape change in this mass region than near N = 88. 5. Calculations 5.1. G R O U N D BAND INTENSITY RATIOS
Because of strong inelastic excitation of rotational states, it is necessary to use a coupled-channel treatment of transfer reactions in deformed nuclei. Tamura et el. 9) have used coupled-channel equations to calculate cross sections for 176yb(p, t) to
(p, t) REACTIONS
257
members of the ground band of 174yb" Ascuitto and co-workers 1o, 12, 13) have used the "source-term" method 84) for (p, t) reactions. This method does not use the exact CCBA equations but it is believed to give the same results 9). The calculations of Tamura et al. 9) agree well with those of Ascuitto et al. lo, 12), so we will discuss only the latter. Ascuitto, Glendenning and Sorensen 1 o, ~2) adopted the Bohr-Mottelson adiabatic hypothesis to write wave functions. The intrinsic wave functions were BCS wave functions constructed from the single-particle eigenfunctions of a deformed WoodsSaxon potential with a shape defined by the deformation parameters f12, f14 and fit. The deformation constants for the protons, tritons and bound neutrons were separately scaled from values obtained from analysis of c~-inelastic scattering 1 s) to ensure the same multipole fields for all particles. The same values of flz were used for both 176yb and 174yb. The BCS calculations were done with the 20 levels nearest the Fermi surface. The strength of the pairing force, G, in the BCS calculations was adjusted to reproduce a gap parameter o f A , = 0.36 MeV for 176yb and 0.575 MeV for 174yb instead of the experimentally 29) observed odd-even mass differences, Pn = 0.593 and 0.687 MeV respectively. Ascuitto et aL 12) suggest that having An < Pn may be necessary due to the low density of single-particle levels in this region and to obtain a smooth transition to the more neutron deficient Yb isotopes. Optical parameters were obtained from fitting inelastic proton data for 176yb from this experiment (sect. 3) and the 20 MeV elastic triton data of Flynn et al. 85) extrapolated to Yb values. The only remaining parameter is a normalization constant which is the same for all states. Fig. 23 shows the fits obtained 12) to preliminary data from this experiment. The CCBA calculations fit the data extremely well, especially when compared to concurrent DWBA calculations which give reasonable agreement with 0 + and 4 + cross sections but badly overestimate 2 + cross sections and underestimate 6 + cross sec, tions. Furthermore the DWBA calculations do not reproduce the 2 + shape well. The critical feature of the calculations is the multi-step processes not included in DWBA but which are explicitly part of the CCBA calculations, For the 2 + state the main two-step processes are (i) inelastic excitation of the target by the incident proton to the ground state band 2 + followed by pickup to the 2 + in the residual nucleus, or (ii) pickup to the residual ground state followed by inelastic excitation to the 2 + by the exiting triton. Ascuitto et aL i o, ~2) show that the strengths of these two-step processes are only slightly less than that of the direct route, that all three routes have angular distributions with shapes different from each other and the data, and that all three routes overestimate the magnitude of the cross section. However, the three processes interfere destructively to give a good fit to the data. 5.2. EXCITED K = 0 BAND INTENSITIES Abdulvagabova and Pyatov l l ) have done PWBA-RPA calculations with ordinary pairing forces to calculate K = 0 bands in the Yb isotopes. They predicted excited
258
M. A. O O T H O U D T
A N D N. M. H I N T Z
0 + states in 174yb at 1.45, 2.00 and 2.58 MeV with 16, 11, and 5 % of the ground state strength in the (p, t) reaction. These states correspond well in strength with the experimentally observed states at 1.489, 1.886 and 2.100 MeV with 26, 12 and 7
,
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Fig. 24. C o u p l e d channel (CC) calculations for t h e K = 0 b a n d calculated at 1.47 M e V a n d experimentally observed at 1.489 MeV. T h e n o r m a l i z a t i o n h a s been determined by fitting the g r o u n d state. Results o f D W B A calculations are also s h o w n . T h e theoretical curves were t a k e n f r o m fig. 8 o f r e f . 13). T h e d a t a points are f r o m this work.
0
60
120
180
OC.M.
Fig. 25. C o u p l e d c h a n n e l calculations for the K = 0 bal~d in ~74Yb calculated at 0.93 M e V a n d experimentally observed at 1.886 MeV. NormalizatiorL is fixed by fitting t h e g r o u n d state. T h e theoretical curves were t a k e n f r o m fig. 9 o f ref. la). T h e d a t a points are f r o m this work.
of the ground state strength. However, for 172yb ' 170yb and 168yb ' states between 1.5 and 3.5 MeV were predicted with 20 to 35 ~ of the ground state strength. Experimentally no such strength is observed UP to 4 MeV of excitation energy. Abdulvagabova and Pyatov 11) suggested that quadrupole forces may be necessary to fit experimental results. Further calculations have been carried out by Ascuitto and Sorensen 13) for excited K -= 0 bands. A more rapid method was used to calculate the form factors
(p, t) REACTIONS
259
than previously 1 o, lz), but in principle these calculations differ from those discussed in subsect. 5.1 only by the addition of an ordinary surface-peaked quadrupole force. The excited K = 0 bands have only a few quasiparticle components, so their energies and strengths are strongly dependent on the energies of the single-particle levels. Due to the lack of accurate information on single-particle energies, Ascuitto and Sorensen ! 3) chose to use levels generated in a Woods-Saxon well using parameters from the lead region. Thus they did not expect to fit either the experimental energies or strengths for individual states, but they did expect to obtain the correct total 0 + strength. The strength of the pairing force was adjusted as in the ground band calculations lo, lz). In order to fix the strength of the quadrupole interaction, the average energies of low lying K = 0 bandheads and the B(E2) and p(E0) values from these bands to the ground band or within these bands were fit. The agreement with the electromagnetic data is good. In the reaction calculations for excited K = 0 bands, the 0 +, 2 + and 4 + members of the target and final K = 0 bands were included in the coupling. Inelastic excitation of the excited K = 0 bands from the ground state was not included, since interband B(E2) values are so small that the inelastic processes would result in a two-step cross section two orders of magnitude weaker than the direct route 13). The calculations reproduce the ratio of the sum of the excited 0 + strength to the ground state for a76yb(p, t) quite well: 46 ~ predicted compared to 45 ~ experimentally. Fits to preliminary data for the 1489 and 1886 keV K = 0 bands in 174yb are shown in figs. 24 and 25; data from ref. 8) have been plotted on these figures. G o o d results were also obtained by Ascuitto and Sorensen 13) for 16°Gd, 174yb and 186W(p, t). The sharp decrease in strength for 174yb(p, t) to excited 0 + states was reproduced well in the calculation. In spite of the success of the Ascuitto-Sorensen calculation in fitting the general trends of excited 0 + strength and angular distributions in ground and excited K = 0 bands, several questions remain. The BCS calculations of Ogle et aL 46), for instance, do not converge near 176yb ' i.e., the level density is so low that the BCS equations have no solution for a reasonable value of G. Thus it is not clear that the wave functions used in refs. 1o, 12, 13) are correct. In addition, it is not clear whether the strong 0 + transitions are caused by gaps in the single-particle spectrum, as discussed in subsect. 4.2.2, or by the transfer properties of individual Nilsson levels. For example, if the transfer amplitudes from different orbitals are unequal, excited 0 + strength can occur even without gaps in the single-particle spectrum s6). To see this, consider a simple mode1 with two active orbitals at the Fermi surface. In the 176yb ground state, denoted I176) g, both levels are occupied. In 174yb, 0 + states can be written as linear combinations of pairs of holes in orbitals 1 and 2: 1174) g = [aA1 + bA2] [176) g, ]174)* = [bA 1 - aA2] [176) g,
260
M . A . O O T H O U D T A N D N. M. I t I N T Z
1 and Ai destroys a zero coupled pair in orbital i. If ai is the L = 0 (p, t) amplitude for orbital i, then the cross section ratio for the two 174yb states is
where
a 2 q-b 2 =
R
--
~*
lalb-%al 2
6 g
[aaa+(12b[ z
When the transfer amplitudes are nearly equal, strong configuration mixing (a ~ b) destroys the excited 0 + strength. If, however, al >> a2 (or the converse) large excited s t r e n g t h can be observed even when a ~ b. For example, if a z = 5a 1 and a ~ b, R = ~. For the pure configuration case (a = 1, b = 0), R = 25. Thus without using gaps in the single-particle spectrum or invoking any special collectivity, we have an excited 0 + state populated by (p, t) with between 0.44 and 25 times the ground state strength. It is desirable to make a more quantitative exploration of the effects of gaps, level ordering and transfer amplitudes on the excited L = 0 strength. We have used a pairing force approximation to calculate 0 + states and the DWBA to calculate (p, t) cross sections for states in 174yb. We did not perform CCBA calculations, since refs. lo, 12, 13) show that DWBA is nearly as good as CCBA for reproducing the shapes and relative intensities of 0 + states. An exact diagonalization of the pairing force was performed with the computer code S P R I N G 87, 88) to avoid problems with BCS non-convergence. The G-values and energies of twelve Nilsson levels for 175yb from ~ [642] through J~-[615] (see fig. 16) were taken from ref. 46). The proper number of pairs for the nucleus considered were placed in these levels in all possible combinations. For most calculations the pairing force was diagonalized only among the 30 states with the lowest summed Nilsson single-particle energies. These 30 states span an energy range of 5 to 6 MeV. The eigenvalues and eigenvectors then give excitation energies for 0 + states and occupation probabilities (V 2) for the Nilsson levels. Spectroscopic amplitudes (X) for (p, t) and (t, p) were then computed from the eigenvectors for the target and residual nucleus. The X were converted from a Nilsson basis to spectroscopic amplitudes in a spherical harmonic basis (S ~) using the Nilsson coefficients 89) for a deformation parameter of t/ = 0.6. The S ~ were used in T W O P A R 87, 9o), a zero-range two-nucleon transfer DWBA code. In the DWBA calculation the single-particle bound-state wave functions were calculated in a Woods-Saxon well with a diffuseness of 0.65 fm, a spin-orbit coupling of 25 times the Thomas term and a radius of 1.25 A ~ fro, where A is one mass unit less than the target. The depth of the well was adjusted for each single-particle wave function to give a binding energy equal to half the two-neutron separation energy in table 2. Note that the S ~ are calculated for a spherical harmonic oscillator well, whereas a spherical Woods-Saxon well is actually used for the reaction calculation. The optical model parameters for the protons were set 5 in table 3. For tritons, the parameters of Flynn et al. 85) for elastic scattering on 182W were modified according to the mass and energy dependence of the Beechetti-Greenlees global triton param-
261
(19, t) REACTIONS eters 91) to Y b values:
vR=
183.1-6.4 (¢-~o)-0.17(E-20)
MeV,
w~=
21.6-110.(4-~o)-0.33(E-20)
MeV,
wo=
0,
FR
1.16 A ~ fro,
a~
q = 1.332 A ~- fro,
= 0.76 fm,
a I = 0.994 fm,
where ~ = (N-Z)/A, 4o is the value o f ~ for 182W and E is the energy o f the outgoing triton for the g r o u n d state o f the reaction considered. TABLE 6
Magnitudes of spherical components a) and relative (p, t) strengths of Nilsson levels li
~[642] ~[523] ~[521]
8
~[6331
9
lh
2g
450
2 499 161
2f
86
35 65 1
54 374
170
26
148
163
~[512] ~[514] ~[624] ½1510] ~[503] ~[512] ~15]
89 586
36 18
8
556 30
55 34 94
9 12 7
238 523
136 654 84
4
530
~n~jb)
1
10
5
22
132
298
515
~rrel.
104
44
½15211
7
3p
3 60 0
488
3d
352
24
14
") Multiplied by 1000. b) Relative cross section at
01a b ~
73
221
13
404 80
469 1000
301
31 32 374 19 522
387 14 13 1000 500 370 11
714
27½°.
Using levels f r o m ref. 46), the energies o f the lowest four 0 + states are calculated to be 0, 1.974, 2.635 artd 2.694 M e V with relative cross sections at 27½ ° (near the L = 0 m a x i m u m ) o f 1.00, 2.33, 0.003 and 0.46 respectively. Experimentally, the excitation energies are 0, 1.489, 1.886 and 2.100 M e V with relative cross sections at 27½ ° o f 1.00, 0.25, 0.12 and 0.05. Thus the predicted 1.974 M e V state contains m o r e than five times as m u c h strength relative to the g r o u n d state as is experimentally seen in the sum o f all excited 0 + states. Since only relative cross sections are calculated, it is n o t clear whether the excited state cross section is too high or the g r o u n d state too low. Calculations using the lowest 56 states in the matrix diagonalization gave no significant improvement 92): excitation energies were shifted by less than 15 keV and the ratio o f the cross section o f the 1.974 M e V state to the g r o u n d state was reduced by less than 10 ~ .
262
M . A . OOTHOUDT AND N. M. HINTZ
To find where the excess strength comes from, the (p, t) strength from each of the Nilsson orbitals and Woods-Saxon single-particle levels was individually calculated. In table 6 the magnitudes of the spherical components t and the relative (p, t) cross section (last column) at 27½° are listed for each Nilsson orbital. Also listed at the bottom of the table (labelled a, zs) are the (p, t) cross sections for each of the spherical single-particle levels relative to the 3d~_. It is interesting that the 3d~ has a higher cross section than the 3p levels. In the lf~ shell the 2p~ level has the highest (p, t) cross section ss). Evidently for this mass region, the 3d~ wave function gives a better overlap than the 3p wave functions with the triton internal wave function in the form factor. Among the Nilsson orbitals, the ½1510] has by far the largest (p, t) cross section due to a large 3p~ component. The cross sections for different Nilsson orbitals vary by a factor of up to 100, easily giving the factor of 5 in amplitudes used in the example above. TABLE 7 Spectroscopic amplitudes for 0 + states populated by 176yb(p, t)lT*Yb E~
Nilsson orbital
(MeV)~[5211 415231 416421 ½15211 ½[633] 415121 -~[5141 3[624] ½15101 315121 -~[5031 ~[615] 0.000 1.974 2.634 2.693
0.052 0.034 0.070 0.042 0.022 0.011 0.082 0.041
0.034 0.041 0.011 0.040
0.086 0.206 0.120 0.776
0.083 0.185 0.092 0.499
0.117 0.936 --0.099 --0.277
0.934 --0.104 --0.048 --0.050
0.286 0.054 0.046 0.025
0.157 0.050 0.009 0.030
0.068 0.080 0.049 0.031 0.035 0.016 0.009 0.005 0.007 0.016 0.021 --0.012
The contributions of the various orbits to the spectroscopic amplitudes for the lowest four 0 + states are shown in table 7. The out of phase component of the 1.974 MeV level, 71514], has too weak a transfer amplitude to reduce the excited 0 + strength sufficiently. In order to reduce the ratio of the excited to ground 0 + cross sections in the example above, the mixing of the two levels must be increased. In the actual case presented here mixing may be increased by either increasing the value of G or by decreasing the gap at the Fermi surface**. Increasing G by a physically unrealistic factor of 2 from 0.132 MeV to 0.264 MeV improves the relative cross sections to only 1.00, 0.50, 0.001 and 0.04. This change is mainly due to an increase by a factor of approximately 4 in the ground state cross section due to an increase in the ½1510] occupation in the target. The excitation energies, however, have become worse: 0, 2.156, 2.865 and 2.984 MeV. The levels above the -~[512] (see fig. 16) were shifted downward together by as much as 400 keV to decrease the gap in the single-particle spectrum between the * The magnitude of a spherical component is defined as the absolute value of S~_ for an (nlj) component in the L = 0 pickup of a pair from a Nilsson orbital with X = I. ,t Cf. the discussion of t h e two-level pairing phase transition model of Broglia et al. 41, 44, 4s) in subsect. 4.2.2.
(p, t) REACTIONS
263
~[512] and the 71514] levels. The best agreement in cross section ratios is obtained for the 400 keV shift, which gives relative intensities of 1.00, 1.76, 0.009 and 0.44 and excitation energies of 0, 1.227, 1.878 and 1.941 MeV. Thus we have not been able to get good agreement with excitation energies and cross section ratios for large changes in either G or the gap in the single-particle spectrum. Furthermore, it may be expected that such large changes will cause disagreements with experimental oddeven mass differences and quasiparticle energies which were fit in ref. 46) to obtain energies for the Nilsson levels used here. It is interesting to note that the quasiparticle energy of the ½1510] orbital is known 46) to no better than 100 keV. Thus it may be possible to shift this level with its large (p, t) amplitude to give the ground state enough strength to dominate the excited states without destroying agreement with quasiparticle energies. Preliminary calculations 92) indicate that the ratio of the first excited 0 + state to the ground state and the excitation energy can be reproduced well by shifting the ½1510] down to slightly above the 71514] level. Further calculations are in progress and will be reported later. We conclude from these calculations that comparison with cross sections and energies for 0 + states excited in (p, t) may be a very sensitive test of proposed singleparticle level schemes. It must be emphasized, however, that the analysis presented here is quite crude. In particular, the effects of ordinary quadrupole and spinquadrupole forces may affect the strength of excited 0 + states 4o).
6. Summary and conclusions The (p, t) reaction has been used to study a number of nuclei in the rare earth region. The reaction has been shown to be good for identifying L = 0 transfers even with incomplete angular distributions. The reaction is not as reliable for determining other L-values. For even-N nuclei many states have been identified through comparison with known level schemes. Members of ground state rotational bands up to 8 + in some cases and 6 + in most have been observed with targets of is°I-If, 17SHf and all the stable even-N Yb isotopes. Calculations presented in the recent literature have shown that DWBA calculations do not fit the excited members of the ground bands nearly as well as do CCBA calculations. The existence of excited K = 0 rotational bands strongly populated by (p, t) has been shown. These bands are observed in 174yb and 176Hf' but are seen only weakly in other nearby nuclei. These bands have been discussed in terms of possible nuclear collective modes such as the pairing phase transition. CCBA calculations 13) with a surface-peaked quadrupole force have been discussed and shown to give a good account of the data. It has also been shown with DWBA calculations and simple pairing theory that a strongly populated excited 0 + state may result from Nilsson singleparticle orbitals with larger than average (p, t) amplitudes. It is not clear whether the strong 0 ÷ states are produced by collective modes, strength from particular
264
M.A. OOTHOUDT AND N. M. HINTZ
Nilsson orbitals, gaps in the single-particle spectra or some combination thereof. Further theoretical investigation of the 0 + states is necessary. A number of other bands have been observed in the even-N Yb and H f nuclei. Members of g a m m a and octupole vibrational bands have been populated with appreciable strength. In addition several previously/known two-quasiparticle states have been observed. It is interesting that several unnatural parity states, such as 3 + members of g a m m a bands, were observed with weak but nonzero cross sections. In view of the problems with calculations for 0 + states, no attempt has been made to do calculations for these states. The (p, t) reaction to states in the odd-N nuclei 171yb and 169yb has also been studied. The lowest L = 0 transfer has been found to proceed to a single-quasiparticle state with the same quantum numbers as the target ground state but with reduced intensity relative to the even-N nuclei. The L = 0 transfer to a state at 1513 keV in 171yb with 13 % of the strength of the lowest L = 0 transfer may indicate the presence of a K = 0 vibration of the even-N core coupled to the unpaired ~[512] neutron. Due to the high density of states excited and greater theoretical complications, considerably less information has been extracted for the odd-N nuclei than for even-N nuclei. Natural targets for elements in the rare earth region have been used to obtain information on trends in cross sections for members of ground bands, g a m m a bandheads, 3 - oetupole states and strongly excited 0 + states. In addition it proved to be rather easy to obtain $2, values that agree well with more ditfi~ult mass measurements. We have shown that the (p, t) reaction is a useful spectroscopic tool in the rare earth region. However, there is a need for more extensive experimental and theoretical studies. The use of Prof. Benjamin F. Bayman's computer codes T W O P A R and S P R I N G and several useful discussions are gratefully acknowledged. We also wish to thank Judy Gursky of the Los Alamos Scientific Laboratory for preparing the H f and Os targets, George Ott for preparing the other targets, Philip Debenham, Albert Kuhfeld, Ralph De!ong and Richard Wallen for aid in data taking and Marjorie Maloney, Mary Evenson and Christie Helgeson for scanning the plates. One of us (M.A.O.) wishes to express his thanks for support received during this work from a N A S A Traineeship (1968-9) and an N S F Graduate Fellowship (1969-72). References
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(P, t) REACTIONS
265
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