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Polarization and angular correlation measurements following the 40Ar(p, nγ)40K reaction
I
2~.B
Nuclear Physics A143 (1970) 481 --496; ~ ) North-Holland Publtshlng Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
POLARIZATION AND ANGULAR CORRELATION MEASUREMENTS FOLLOWING THE *°Ar(p, ny)4°K REACTION P. J. TWIN t, W. C. OLSEN and D. M. SHEPPARD
Nuclear Research Centre University of Alberta, Edmonton, Canada tt Received 12 November 1969 Abstract: Gamma-ray angular distributions and polarization correlations have been measured using the reaction 4°At(p, ny)4°K. All the experiments were carried out with bombarding energies close to threshold for the states under investigation. Comparison with compound nuclear predictions has yielded the following spin assignments to states in 4°K; 1.959(2+), 2.047(2-), 2.070(3-), 2.103(1 -), 2.261 (3+), 2.290(1 +), 2.291 (4, (3)), 2.419(2,3), 2.575(2,4) and 2.625(0-). Multiple mixing ratios together with the gamma-ray branching ratios have been determined for all these levels. A short theoretical treatment of the polarization of gamma rays from aligned nuclei using the phase-consistent formalism of Brink and Rose is included. E[
N U C L E A R REACTIONS 4eAr(p, n~'), E = 3-2--5.2 MeV; measured ] tr(E; ET, 0~,),y-polarization. 4OK deduced levels, J, ~, y-branching, ~-mixing. Natural target.
1. Introduction
Recent work by Gerace and Green l) and Flowers and Skouras 2) has focussed attention on the positive parity states around 4°Ca. These states are assumed to be mixtures of 2p-2h, 4p-4h and even 8p-8h configurations. Similar particle-hole states with positive parity should exist in 4°K. We have used the 4°Ar(p, ny)4°K reaction to search for such states and have definitely identified four positive parity states in 4°K at 1.644 MeV (0+), 1.959 MeV (2+), 2.261 MeV (3 +) and 2.290 MeV (1+). Previous work on 4°K, summarized by Endt and van der Leun 3), included the 39K(d, p)4°K experiments of Enge et aL 4). They found the four lowest states at 0, 0.030, 0.800 and 0.891 MeV, were strongly excited via In = 3 transfers, indicating these states were predominately single p-h states of the configuration fld~'t, and hence had negative parity. The states at 2.047, 2.070, 2.103 and 2.625 MeV were also strongly excited but were characterised by In = 1 transfers and therefore were ascribed as negative parity states arising from the p~d~ 1 configuration. All other states they observed below 2.6 MeV had only a very weak excitation cross section in the (d, p) reaction. However, Main et al. 5) reported that the 1.644 MeV state is strongly excited in the 4°Ar(p, n)4°K reaction and measured its lifeti~ae for y-decay to be 1" On leave of absence from the University of Liverpool, England. tt Work supported in part by the Atomic Energy Control Board of Canada. 481
482
P.J. TWINet al.
0.49 Its. They observed the state decayed to both the 0.800 MeV (2-) and 0.030 MeV (3-) states and concluded its spin was probably 0 ÷ ; a 1 ÷ assignment, however, could not be excluded. The 4°Ar(z, t)4°K reaction has recently been investigated by Wesolowski et al. 6) who observed that both the 1.644 and 1.959 MeV states (the lowest states not ascribed to single p-h configurations) were strongly excited. Using a DWBA analysis, they suggest a negative parity assignment for the 1.644 MeV state with J < 2 (inconsistent, however, with the lifetime reported by Main) and positive parity for the 1.959 MeV state with a possible spin of 3. We recently reported 7) measurements of the neutron angular distributions of the low-lying states in 4°K following the 4°Ar(p, n) reaction showing that the 1.644 MeV state has indeed a spin of 0. The report also included a summary of the results of ~-ray angular distribution and polarization experiments identifying three positive parity states in 4°K. The techniques used together with the results for all the lowlying levels in 4°K are described in the present paper.
2. Experimental techniques The proton beam from the University of Alberta Van de Graaff accelerator was collimated to 0.15 cm diameter 15 cm before striking the target gas cell. This cell consisted of two tantalum hemispheres of radii 1.5 cm and l.9 cm with a 7.7 mg ~cm- 2 thick platinum window placed centrally in the hemisphere of larger radius. Thus the beam passed through 0.6 cm of natural argon gas maintained at a pressure of one atmosphere (a 65 keV thick target to 5 MeV protons). For all angular distribution experiments, the system isotropy was checked using the strong 1614 keV v-ray from the decay of the 1.644 MeV (0 ÷) state in 4°K. This ~-ray was observed at all proton energies above 4.3 MeV. 2.1. GAMMA-RAY ANGULAR DISTRIBUTIONS The Ge(Li) detector used in this work had an active volume of 45 cm 3 with a resolution of 3.5 keV for 1332 keV and 4.5 keV for 2615 keV ?-rays. Spectra were recorded at angles of 0, 30, 45, 60 and 90, each angle being repeated at least once. Above a bombarding energy of 4.5 MeV, the intensity of the 1614 keV ?-ray was used as a monitor whilst below this energy the total yield in a NaI crystal was used. The spectra were analysed with a peak-fitting program written by Tepel s) to give the total number of counts in the full energy peaks. The resulting angular distributions were fitted in terms of even-order Legendre polynomials. The a 2 and a 4 coefficients were then compared with the predictions for various spin sequences and multipole mixing ratios, of the compound nuclear statistical model. We used the M A N D ¥ program of Sheldon and Van Patter 9) which uses the Biedenharn 1o) convention for the sign of the mixing ratio c5. This yields the opposite sign for dipole-quadrupole mixing from the Rose and Brink 11) phase-consistent formalism which is used throughout this paper. Neutron and proton transmission coefficients,
'*°ar(P, m')"
483
required by the M A N D Y program, were calculated from the average optical-model parameters of Perey and Buck as tabulated by Rosen 12). The calculated angular distributions were found to be sensitive to the ratio of the l = 1 and I = 0 transmission coefficients for the outgoing neutrons. To minimize this effect, the bombarding energy was kept within 100 keV of threshold so that the magnitude of the I = 1 transmission coefficient was less than 10 ~ that of the l = 0 transmission coefficient. To obtain a quantitative comparison between the compound nuclear predictions and the experimental data, a Z2 analysis using a grid search program was carried out in terms of the initial state's population parameters. These were limited to lie within values calculated at threshold and 200 keV above threshold by the M A N D Y program. This technique enabled angular distributions of all primary T-rays from a state and also 7-T correlations and polarization data to be analysed simultaneously. 2.2. G A M M A - G A M M A A N G U L A R CORRELATIONS
Two 7.6 cm by 7.6 cm NaI crystals, placed 10 cm from the target, were used for the correlation measurements. One was fixed at 90 ° relative to the beam direction whilst the other was moved over an octant such that the angle relative to the beam direction varied from 0 ° to 90 ° and the angle between the planes of each detector and the incident beam varied from 90 ° to 180°. A conventional fast-slow coincidence system was used together with the sum coincidence technique. The total yield of T-rays in the fixed detector was used as a monitor and at least 2 runs were completed at each angle. ANALYSING CRYSTAL INCIDENT PARTICLE
ELECTRIC V E C T O R
TARGET
~
INCIDENT J " GAMMA RAY
SCATTERED GAMMA
f
SCATTERERI \ ANALYSING CRYSTAL
Fig. 1. A schematic representation of the Compton polarimeter indicates the positions of the target, the scattering and the analysing crystals and the electric vector of the ),-ray, together with the angles discussed in the text.
The grid search program (subsect. 2.1) was used to fit the correlation in terms of the population parameters of the initial state. These were again limited to lie within the compound nuclear predictions at threshold and 200 keV above threshold. "Ihe program was based on the tables of Smith 1 3 ) i n which only quadrupole-dipole mixing in the T-rays is considered. ,
484
P.J. TWIN e t al.
2.3. G A M M A - R A Y P O L A R I Z A T I O N M E A S U R E M E N T S
The polarizations of the y-rays were measured by a Compton polarimeter shown schematically in fig. 1. The scattering crystal was a 45 cm a Ge(Li) detector placed 20 cm from the target. The analysing crystals were 7.6 cm diameter by 7.6 cm long NaI detectors placed 12 cm from the scattering crystal and making an angle with respect to the line from the target through the centre of the scattering crystal of = 82 °. One crystal was at ~b = 90 ° and the other at ¢ = 0 ° with respect to the reaction plane. Both crystals were shielded by 15 cm of lead from direct observation of y-rays originating in the target. The complete polarimeter could be moved from 0 = 90 ° to 0 = 0 ° relative to the beam direction. A conventional fast-slow coincidence system was used together with the sum coincidence technique. This resulted in two spectra, as shown in fig. 4, one associated with the scatterer and the 0 ° analyser, and the other associated with the scatterer and the 90 ° analyser. The yields in the full energy peaks, N(O, ¢) were obtained from the spectra. The small difference in the efficiencies of the analysing crystals was corrected for by normalising the yields at 0 = 0 °, where the y-rays are unpolarized. We now present a treatment for the theoretical calculation of the polarization based on the formalism of Rose and Brink 1i). Previous articles by Fagg and Hanna 14) McCallum is), Suffert et al. 16) and Ferguson 17) use various formulations of angular momentum coefficients and polarization definitions. The angular distribution of a 7-ray measured by a detector, insensitive to polarization, from an aligned state of spin J1 with magnetic substrate populations P(M1 ) to a state ']2, M 2 is given by
W(O) = Z P(M1)pk(J1 M1)6"(1 + 62) - 1Rk(LL'J1Sz)QkVk(COS0),
(1)
kM t LL"
where LL' represents the multipolarities (L' _~ L) of the transition, n is 0, 1 and 2 for (LtLI), ( L I L I + I ) and ( L I + I L I + I ) radiation respectively and the Pk and Rk terms are defined and tabulated by Rose and Brink 11). The direction polarization correlation is given by
W(O, T) = •
P(MI)Pk(JI M1)J"(1 +62) - 1Rk(LEJ1 J2)Qk
kMjLL"
x [Pk(cos 0 ) + ( - - ) " COS 2VKk(LE)p2(cos 0)],
(2)
where V is the angle between the reaction plane and the electric vector of the emitted y-ray. The Kk(LL') coefficients are listed by Fagg and Hanna 14) and repeated in table 1 for convenience. If the L' radiation is electric in character then n' = 0 and if L' is magnetic n' = 1. A convenient definition of the theoretical polarization is Pt =
W(O, V = 0°)- W(O, V = 90 °) W(O, V = 0°) + W(O, V = 90°)
which limits Pt to values between + 1 and - 1 with Pt T-ray.
=
(3)
0 indicating an unpolarized
'*°Ar(p,ny)
485
Experimentally the yields in the full energy peaks of the sum coincidence spectra N(O, ~b = 0) and N(O, ~ --- 90) were measured. These correspond in the scattering crystal to experimental Compton scattering intensities for ~, --- 82 ° in the reaction plane and perpendicular to the reaction plane respectively. Hence, neglecting solid angle effects, N(O, c~ = O) = W(O, 7 = 0 ) d a o + W(O, "t = 90)da9o, N(O, q5 = 90) = W(O, 7 = O)da9o+ W(O, y = 90)dao, where dao and do'9o are respectively, the differential cross sections for Compton scattering, at ~, = 82, in and perpendicular to the plane formed by the electric vector and the direction of the incident y-ray. TABLE
l
Numerical values of the coefficientsKk(LL') LL"
k =2
k =4
12 22
-~ ~-
-- -1 2~
Therefore an experimental value for the polarization can be obtained from pg
~---
--
_
1 ,v(o,
= o)-N(o,
= 90)
R N(O, 4) = O) + N(O, ~b = 90)' where, R - da9°-da° do'9o + dtro Hence, using the Klein-Nishina formula R-
sin 2 ~,
ko/k + k/ko - sin2~, '
where k and ko are the wave numbers of the incoming and scattering radiation and ¢, the angle through which the ~,-ray is Compton scattered. R is essentially the polarization efficiency of the polarimeter and can be determined experimentally using y-rays with known multipole mixing ratios. However, Ferguson 17) has suggested a technique for calculating R which takes account of the solid angles subtended by the analysing crystals. At a y-ray energy of 1.5 MeV, the value of R calculated by this method was checked to within 5 ~o using angular distribution and polarization measurements for the 1460 keV 2 + --* 0 + transition in 4°Ar. In the present work, we use these calculated values of R, assuming a nominal 10 ~o error, in determining
P. $. T W I N e t al.
486
polarizations from the experimental N(O, q~) yields. This data was then fitted simultaneously with the angular distributions using the grid search program (subsect. 2.1) which is based on formulas (1) ~ (3) above. The population parameters were restricted to lie within compound nuclear predictions at threshold and 200 keV above threshold. We now show how an estimate of the expected theoretical polarization at 0 = 90 ° can be obtained in terms of the experimentally measured a2 and a4 Legendre coefficients fitted to the angular distribution of the "/-ray. If one considers that only dipole and/or quadrupole de-excitation radiation occurs, then the theoretical polarization measured at 0 = 90 ° taken the following form
E Q2P(M1)P2(J1M1){3[R2(llJ1J2)-F2r~R2(12J1J2)+r~2R2(22J1J2)] MI
-- 8fig2(12J1 J2)} + 1.25 pt =
-I
~ Q4P(M1)p4(JiM1)62R4(E2J1
J2)
MI
-- 2 E P(M1)po(J1M1)(1+62) -
E Q2 P(ML)p2(J1M,)[R2(llJ1J2)
MI
MI
+ 2~R2(12JiJ2) +
62R(22J1J2)]+ 0.75
~
Q4P(M1)P4(JIM1)62R4(22JiJ2)
MI
with the positive sign applicable to MI/E2 mixing and the negative sign to El/M2 mixing. The term
E P(M1)po(J1M,) --- 1
MI
and the terms Ml
Q2P(M~)p2(JIMt)[R2(11J1J2)q-26R2(12J~ J2)+ 52R2(22J1 E Q,~P(M1)p2(J1M~)fi2R4(22J1 Jz)/(1 + fi2),
J2)]/( 1 q- 5 2 ) ,
MI
respectively, are in fact the theoretically predicted a 2 and a4 Legendre polynomial coefficients of the angular distribution of the gamma ray, so that 3a 2 - 8 ~ Q2 P(M1)P2(J1Ml)g2(12Jt J2)6/(1 + 62) + 1.25a4 Pt
=
--
M~
2 - a 2 + 0.75a4 A further simplification is obtained when only pure dipole or quadrupole radiation is present: 3az + magnetic dipole: Pt = - + - - 2- a2 - - electric quadrupole:
Pt = -+"
3a2 + 1.25a4 2 - a 2 + 0.75a4
+ electric -magnetic"
These a2 and a 4 coefficients can then be replaced by their experimental estimates obtained from an analysis of the measured angular distribution. Hence a reasonable
4°At(p, nr)
487
prediction of the theoretical polarization for various spin and parity assignments can be obtained. 3. R ~
The experimentally measured branching ratios and Legendre coefficients together with the spin-parity assignments and associated ~-ray multipole mixing ratios are listed in table 2. The energies of the levels, quoted in the table, have been determined TABLE 2 The results listed are for the states of 4 ° K below 2.65 MeV excited close to the threshold for excitation. The energies of the states have been determined from y-ray measurements in the present work a n d have an error of less than 1.5 keV. The branching ratios have been rounded and have an error 4- 3% Initial state (MeV)
Final state (MeV)
Branching ratio
0.800 1.644
2.291
0.030 0.800 0.030 0.800 0.030 0.800 0.030 0.~0 0.800 0.030 0.000 0.800 0.030 0.030 0.000 1.644 0.800 0.000
100 20 80 85 15 40 30 30 10 55 35 30 70 80 20 65 35 100
2.397
0.030
(50)
2.419
0.000 0.800
(50) 90
2.575
0.030 0.000 0.030
5 5 100
2.625
2.103
100
1.959 2.047
2.070
2.103 2.261 2.290
Legendre coefficients a2 a.
Initial spin
--0.094-0.01
0.004-0.01
20+
0.404-0.01 --0.234-0.01 0.484-0.07 --0.12i0.01 0.16--0.03 --0.6910.10 0.61±0.04 --0.03 4-0.04 --0.124-0.01 0.01 4-0.02 0.414-0.01 --0.10::t=0.02 --0.154-0.03 --0.~4-0.03 0.55-/-0.04
0.00±0.01 0.004-0.02 0.074-0.07 --0.01 i 0 . 0 1 0.00t0.04 0.094-0.12 --0.01±0.05 0.01 4-0.05 0.004-0.01 0.01 +0.02 --0.01 +0.01 0.024-0.03 0.064-0.04 --0.014-0.05 --0.044-0.05
2+ 2-
3-
13÷ 1÷ 4(3)
Final spin 3232323423423340+ 24-
Multipole mixing ratio ¢5 0.004-0.01 M2 E3 0.004-0.02 --0.104-0.04 --0.13+0.09 --0.01 4-0.02 E2 0.204-0.10 --0.274-0.10 0.07 4-0.05 --0.30-t-0.06 E2 --0.024-0.05 0.044-0.06 MI --oo < ~ < --0.3 --0.354-0.25(4)
1.o ±0.3 (3) 30.344-0.03
--0.21 =t:0.03
0.01 4-0.04
0.06-'-0.04
2,3
42-
2,4
343-
0-
1-
--2.0-t-0.6 (2) -0.05--'O.lO(2) -0.354-0.05(3)
--0.08 --0.03(2) --0.06 ~0.02(4) M1
from the measurement of 7-ray energies using the Ge(Li) detector with a 56Co spectrum being taken simultaneously to ensure an accurate calibration. The energies quoted in table 2 have an uncertainty of less than 1.5 keV. The following detailed
488
P.J. TWXNet al.
discussion of the results mentions only a2 Legendre coefficients because none of the angular distributions yielded an appreciable a4 coefficient. 3.1. THE 0.800 MeV STATE
The spin of this state was known to be 2 - and the decay to proceed entirely via a 770 keV y-ray to the 0.030 MeV ( 3 - ) state. However, the E2/M1 multipole mixing d
{o)
1OOOO
100
(b)
100~
X2 100
10 CONFIDENCE LIMIT
:ONFIDENC LIMIT
-90 -45
0
arctan
45
90
lJ
,
-90 -45
,
i
i
0
45
90
arctan
Fig. 2 a) The Zz plot for the 770 keV y-ray angular distribution, measured close to threshold, fitted in terms of the compound nuclear model. b) The 7.2 plot for the angular correlation between the 770 keV and 844 keV y-rays from the 1.644 (0 +) state.
ratio of this y-ray was undetermined. The angular distribution measured at Ep = 3.23 MeV within 40 keV of threshold yielded an a2 of - 0 . 0 9 + 0.01 which could be fitted with two values of 6 as shown by the X2 plot in fig. 2a. To resolve this ambiguity, the Y-7 correlation between the 770 keV and 844 keV -:-rays from the decay of the 1.644 MeV (0 +) state was measured. As the population parameters of the initial state are uniquely determined and the 844 keV 7-ray must be pure quadrupole, the correlation is sensitive only to the mixing ratio of the 770 keV y-ray. The X z plot for the correlation shown in fig. 2b enables the previous ambiguity to be resolved and the 770 keV ),-ray shown to be pure MI. 3.2. THE 1.959 MeV STATE
This state decays 85 Y/ovia a 1159 keV y-ray to the 0.800 MeV ( 2 - ) state and 15 ~o via a 1929 keV y-ray to the 0.030 MeV (3-)state. The angular distributions of both
4°~Lo, m,)
489
~sK 1159 key
~S~~~~~ .),-RAY o') i1
1000
X2
....~-
~J
Z IOK 0U 2 3 4
11[,
,~ 0.1 I[ --H-CONFIDENCEIl LIMIT
1.959 - ~
°'"°°-'-l-r
z
0030"--z-3" "OK 1.OK
~//
y-RAY
I
L0 nrctan ~1159
I
015 cos 2 8
0
Fig. 3. T h e 1.959 M e V state: the Z z plot for the s i m u l t a n e o u s fit o f the a n g u l a r distributions o f t h e 1159 keV a n d 1929 keV 7-rays, m e a s u r e d at a b o m b a r d i n g energy o f 4.44 MeV, 90 keV a b o v e threshold. T h e experimental a n g u l a r distributions together with the best fits for spin a s s i g n m e n t s o f 1, 2, 3 a n d 4 are shown.
(a)
4Kj3!
z
2Kt,0Aa
O
1KF v . ~ . . j ~11,726
I
i~1460 0° ANALYSING II '°A CRYSTAL n59
ol
[ ,,n 3KI Z
(b)
II 1614
1.o
II A
v ~
,0A
Z
~
o
A,.o
t
]J4"OA- 90" ANALYSING II CRYSTAL II 1614
110~3/~
I I It
37Ci
I Kl'-''',--J ~It1,26 ol " ~ CHANNEL
,o~ 2s23
0.5 a=0 12-
, oo
N ,v <~
-0.5
a.
-1.0 9~0°
4'5"
Ir 0
N°.
Fig. 4. T h e 1.959 M e V state a) T h e s u m coincidence spectra at a b o m b a r d i n g energy o f 4.44 MeV, 90 keV above t h r e s h o l d for the 1159 keV y-ray f r o m the 1.959 M e V state, are s h o w n for both 0 ° a n d 90 ° analysing crystals. b) T h e m e a s u r e d polarizations o f t h e 1159 keV y - r a y at angles o f 90 ° a n d 45 ° to the b e a m direction are s h o w n together with the best fits at t h e two angles for spin a s s i g n m e n t s o f 2 a n d 3 when the data was fitted s i m u l t a n e o u s l y with the a n g u l a r distributions o f the I 159 keV a n d 1929 keV y-rays. T h e a n g u l a r distribution data h a d previously indicated the spin o f the state is 2 a n d hence the polarization s h o w s it has positive parity.
490
P. ~. T W I N e t al.
primary ),-rays were measured at a bombarding energy of 4.44 MeV, 90 keV above threshold. They were fitted simultaneously in terms of the 1.959 MeV state's population parameters which were limited to lie within the compound nuclear predictions at threshold and 200 keV above threshold. The resulting Z2 plot shown in fig. 3 clearly indicates that the 1.959 MeV state has a spin of 2 and the 1159 keV 7-ray is pure dipole. The plot shows for each value of 8~59 the minimum X2 obtained when ~1929 is varied from +oo to - o o . Fixing 61159 as zero enables the determination of (~1929 as - 0 . 1 0 + 0 . 0 4 . The y-), correlation between the 770 keV and 1159 keV ),-rays was also measured. However, the correlation yielded no additional information to that obtained from the simple angular distribution of the 1159 keV ),-ray. The explanation is that the only large coefficient in the associated Legendre polynomial expansion of the theoretical ),-)' correlation for this spin sequence is identical with the a 2 Legendre term determined in the simpler experiment. The polarization of the 1159 keV y-ray was measured at angles of 90 ° and 45 ° with the same bombarding energy of 4.44 MeV. The sum coincidence spectra obtained are shown in fig. 4. This data was then fitted simultaneously with the angular distributions of the 1159 and 1929 keV 7-rays. The previous spin assignment o f J = 2 together with the predicted and measured polarizations shown in fig. 4 indicate the 1.959 MeV state must have positive parity. These results confirm the tentative positive parity assignment of Wesolowski 6) bFt show their suggestion of 3 for the spin of the state was incQrrect. 3.3. TI-I~ 2.047, 2.070 AND 2.103 MeV STATES These three states are very strongly excited in the 39K(d, p)4°K reaction with In = 1 transfers and hence were assigned as arising from the p,~d~-1 configuration 4). Four states with spins 0 - , 1-, 2- and 3 - are expected from this configuration and Enge 4) assigned the 2.047 MeV as 3 - , 2.070 MeV as 2 - and 2.103 MeV as l (with the 0 - state at 2.625 MeV) on the basis of excitation cross section. In the present work all three states were strongly excited and their decay modes accurately determined. The 2.047 and 2.070 MeV levels have decay branches to each of 3 lowest states in 4°K whilst the 2.103 MeV state decays only to the 0.030 MeV ( 3 - ) and 0.800 MeV ( 2 - ) states. Angular distributions for all the primary y-rays of the decays of the three levels were obtained at a bombarding energy of 4.55 MeV, between 50 and 100 keV above threshold. The experimental a2 and a4 Legendre coefficients are shown in fig. 5 together with the compound nuclear predictions for final state spins of 2 - , 3- and 4 - . The figure shows that the large a2 coefficients of - 0 . 6 9 40.10 and 0.61 +0.04 for the 1270 and 2040 keV ),-rays indicate the 2.070 MeV state must have a spin of 3. Also the large a2 of 0.48__+0.07 for the 1247 keV ),-ray together with the small a2 of -0.12-t-0.0l for the 2017 keV 7-ray are consistent with a spin of 2 for the 2.047 MeV state, but it is doubtful if a spin assignment of 3 could be completely ruled out. We, therefore, accept the strong evidence that these states do arise
*°Ar(p, n~,)
491
from the pg.d~ ~ configuration and hence assign the 2070 keV state as 3- and the 2.047 1V[eV state as 2-. Furthermore, the anisotropy of the 1303 keV ~. ray indicates the 2.103 MeV state must be the 1- member of this quadruplet. 3.4. T H E 2.261 MeV STATE
This state decays 80 ~ to the 0.030 MeV ( 3 - ) state and 20 % to the 4 - ground state. A possible 1461 keV transition to the 0.800 MeV ( 2 - ) state is completely obscured by the strong 1460 keV v-ray from 4°Ar. Hence a coincidence experiment was carried out with a NaI crystal and the 45 cm 3 Ge(Li). No evidence was found for a 1461 keV 7-ray in coincidence with the 770 keV -?-ray enabling an upper limit of 5 % to be placed on a possible branch to the 0.800 MeV state. PREDICTED 02 A N D 04 LEGENDRE COEFFICIENTS PLOTTED VERSUS B FOR FINAL STATES 2, 3-, 4-
~4 li
---T---J __L_ 4.
-=-.....PURE Q U A D R U P O L E . . . . . . . PURE DIPOLE :.XPERIMEN TAL COEFFICIENTS ~J--- ),-RAYS F R O M 2.047 M e V LEVEL [] .... z-RAYS F R O M 2.070 M e V LEVEL 11ri- -
),-RAYS F R O M 2.103 M e V LEVEL
- 0.8
a4
_J_. J
-0.4
d-----o."-4-- ---o78---*
a4
"~
1-0.8
.a--a--
I
Ir
tI -0.4
1"-0.4
I
°~
J 2-
r
JI
Fig. 5. The a2 and a , Legendre coefficients, predicted by the c o m p o u n d nuclear model, are shown for final state spins o f 2 - , 3 - and 4 - . The multipole mixing ratio 6 varies around the ellipses. The cxperimentally determined coefficients for all the ~,-rays from the 2.047, 2.070 and 2.103 MeV states are plotted.
Angular distributions of the 2231 and 2261 keV 7-rays were measured at a bombarding energy of 4.73 MeV, 70 keV above threshold. When compared with the compound nuclear predictions shown in fig. 5 for final states of 3- and 4 - , the a2 coefficients of 0.41 +_0.01 and - 0 . 1 0 + 0 . 0 2 for the 2231 and 2261 keV 7-rays are only consistent with a spin assignment of 3 for the 2.261 MeV state. The parity of the state was determined by measuring the polarization of the 2231 keV y-ray using the Comp-
492
P.J. Twin
et al.
ton polarimeter which just resolved this ~-ray from the weaker 2261 keV ~-ray. Using the calculated polarimeter efficiency o f 0 . 2 1 + 0 . 0 2 the polarization was determined to be -0.57__+0.30. This data was then fitted together with the 2231 and 2261 keV ~-ray angular distributions. The X2 plot, shown in fig. 6a, clearly establishes the assignment o f 3 + for the 2.261 MeV state. The data further indicates that both the 2231 and 2261 keV 7-rays are predominately E1 transitions. The angular distributions and the polarization measurements together with the best fits for the various spin and parity assignments are shown in fig. 6b. This figure indicates that, in the present case, (b) .~ .~2
2261 keV y-RAY
1o)
1000
0 6K/ , U 1.0
": ::..
•
~6x
44
,\
~.
i
, f 0.5 ¢0S26* 0.0 • 2231 keY ),-RAY
100
L.I
X2
3K
io
~!o o1,
II. o!o •
°3°
1
p .~
"m
ols
COS2
, , . . . . . -I "Or.5 0 ~5 1 2 82231 keV
~
4*
POLARIZATION 0.5| ~ 2 3 " OF 2231 keV )'-RAY O0I"
Fig. 6. The 2.261 MeV state: a) The Z2 plot versus ~ of the 2231 keV),-ray for various spin and parity assignments is shown. The polarization data for the 2231 keV 7'-ray has been fitted together with the angular distribution data for both the 2231 and 2261 keV 7-rays. b) The best fits to the data for the various spin and parity assignments are shown. The fits to the angular distributions are identical for both parity assignments. The results indicate the 2.261 MeV state has a spin of 3 +. the polarization data alone can be equally well fitted by spin assignments o f 2 - , 3 + or 4 - . Hence the spin o f the state must be previously established by the angular distribution data before the polarization measurement enables the parity of the state to be deduced. 3.5. TIlE 2.290 AND 2.291 MeV STATES One state at 2.290 MeV was seen in the 39K(d, p ) 4 ° K reaction. At a b o m b a r d i n g energy o f 4.73 MeV, 40 keV above threshold, we observed possible 7-decay branches to the 1.644 MeV (0+), 0.800 MeV ( 2 - ) , 0.030 MeV ( 3 - ) and g r o u n d ( 4 - ) states, suggesting that the 2.290 MeV state was a doublet. This conclusion was confirmed by
4°At(p, ny)
" 493
j ~
t,O I-Z
~i~
5K
------
-"1 0 U
2*
X 2 • 60.8
2"
X 2,51~4
ca
Z
1÷
X 2 • 0.9
- - - - - - 1-
X 2 " 3.1
5K
0 10
0.5
0
cos 2 0
Fig. 7. The 2.290 MeV state: The angular distribution of the 646 keV 7-ray to the 1.644 MeV (0 +) state is shown together with the best fits for spin assignments 1 +, 1 - , 2 + and 2 - . The state definitely has spin a of I and the fits further indicate that 1 + is the most probable assignment. (b)
|a)
N ~
1000
a2 ,055 ± 0.04
u~ 2500 Z 0 U u_ 2000 0
X 2 100
10 :~ 1500 7
1000
-4;
; arctan
,;
9'0
1.0
05 cos 2 O
0
Fig. 8. The 2.291 MeV state: a) The Z2 plot is shown for various spin assignments to the 2.291 MeV state when the angular distribution of the 7-ray to the 4 - ground state is fitted in terms of the compound nuclear predictions. b) The best fits for spin assignments 3 and 4 are shown together with the experimental angular distribution. These results indicate a spin of 4 for the 2.291 MeV state, although an assignment of 3 cannot be ruled out at the 0.1 ~ confidence limit. the analysis o f the angular distributions, the accurate m e a s u r e m e n t o f the y-ray energies a n d f r o m the o b s e r v a t i o n that the relative intensities o f the four possible
P . J . T W I N et aL
494
decay branches changed at different bombarding energies. From the latter we conclude the following; the 1490 keV ?-ray (to 0.800 MeV state) is associated with the 646 keV ?-ray (to 1.644 MeV state) in the decay of the 2.290 MeV state; the 2291 keV ?-ray is the only predominate decay mode of the 2.291 MeV state; the 2261 keV ?-ray is predominately associated with the 2.261 MeV state and is not a decay from the states at 2.290 and 2.291 MeV to the 0.030 MeV state. The angular distribution of the 646 keV y-ray to the 1.644 MeV (0 ÷) state is shown in fig. 7 together with the best fits for spin assignments of 1 and 2. The population parameters were limited to lie between values calculated at threshold and 200 keV above threshold by the M A N D Y program. These limits are noticeably different for ---1---2.419 2- ' 0.800 / ~ .
/\
40 K
I
-90
I
-45
I
I
I
0
45
90
arctan
Fig. 9. The 2.419 MeV state: The %z plot is shown for various possible spin assignments when the angular distribution ofthe 1619 keVT-rayto the 0.800 MeV (2-) state is compared with the compound nuclear model predictions. the two parity assignments in the case of spin values 1 and 2. Hence fig. 8 shows not only that the 2.290 MeV state must have spin 1 but also that a positive parity assignment is much more likely than a negative parity assignment. The angular distribution of the 2291 keV "t-ray is very anisotropic yielding an az of 0.55__0.04. The Z z plot, shown in fig. 8, for the comparison of the angular distribution with the compound nuclear predictions indicates the 2.291 MeV has a spin of 4, and the multpole mixing ratio of the transition is -0.35___0.25. However, a spin assignment of 3, with a 6 of around unity cannot be completely ruled out at the 0.1 ~o confidence limit.
'*°Ar(p,nT)
495
3.6. THE STATES ABOVE 2.3 MeV The 2.397 MeV state was very weakly excited in the 4°Ar(p, n)4°K reaction. The decay modes observed were approximately equal branches to the 0.030 MeV and the ground state. The 2.419 MeV state predominately decays to the 0.800 MeV ( 2 - ) state with a 1619 keV 7-ray, with weak branches of 5 ~ to both the 0.030 MeV (3 - ) level and the 4 ground state. The angular distribution of the 1619 keV y-ray was measured at a bombarding energy of 4.9 MeV, 70 keV above threshold. The a2 coefficient of 0.34 is consistent with a spin of 2 but as shown in fig. 9, an assignment of 3 cannot be completely ruled out. The 2.575 MeV state is strongly excited and decays entirely to the 0.030 MeV ( 3 - ) state with a 2545 keV ~-ray. The angular distribution of this ?-ray yielded an a2 of - 0 . 2 1 + 0 . 0 3 which is seen from fig. 5 to be consistent with a spin assignment of either 2 or 4. In both cases, the multipole mixing ratio of the ~-ray is close to zero. The 2.625 MeV state was observed to decay entirely to the 2.103 MeV ( 1 - ) state with a 522 keV 7-ray. This adds further weight to its previous identification as the 0 member of the p,ld~- 1 quadruplet due to the strong In = 1 transfer to the state in the 39K(d, p)40 K reaction.
4. Conclusions 4.1. THE NEGATIVE PARITY STATES The only transition between the four lowest fld~-1 states for which we were able to determine c5 is the 770 keV (2- ~ 3 - ) y-ray which is pure M1. This complements previous measurements 3) which show the 30keV ( 3 - - - , 4 - ) y-ray is also M1. We determined the multipole mixing ratios of all 8 transitions between the p~d~. 1 states and the f~d~"~ states. Two of them, the 2047 keV (2- --* 4 - ) and 2073 keV (1- ~ 3 - ) are E2 transitions. The other transitions from the 1- and 2 - states show considerable E2/M1 mixing, see table 2, except for the 2017 keV (2- --, 3 - ) y-ray. All 3 transitions from the 3 - state have large quadrupole components, the smallest being for the 2070 keV (3- ~ 4 - ) y-ray. 4.2. THE POSITIVE PARITY STATES We have established the presence of four positive parity states in 4°K with spins 0 +, 1 +, 2 + and 3 +. They lie at approximately the same excitation energy as the single p-h p~,d~"1 states. Their configuration is probably f~d~-2 or f~d~- 4. However, at present none of the theoretical calculations on p-h states around mass .4 = 40 have been applied to this nucleus. We wish to thank Professor Sheldon for the M A N D Y program and T. P. G. Carola, W. Chung, A. A. Pilt and E. Wong for assistance with the experiment.
496
P.J. TWIN et al.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
W. J. Gerace and A. M. Green, Nucl. Phys. 93 (1967) 110; A123 (1969) 241 B. H. Flowers and L. D. Skouras, Nucl. Phys. A l l 6 (1968) 529 P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 H. A. Enge, E. J. Irwin and D. I-L Weaner, Phys. Rev. 115 (1959) 949 I. G. Main, N. Dawson, D. Kewley, N. Lawley, M. F. Thomas and P. J. Twin, Phys. Lett. 26B (1968) 295 J. J. Wesolowski, L. F. I-Jansen and M.L. Stelts, Phys. Rev. 172 (1968) 1072 P. J. Twin, W. C. Olsen and E. Wong, Phys. Lett. 29B (1969) 570 J. W. Tepel, Nucl. Instr. 40 (1966) 100 E. Sheldon and D. M. Van Patter, Rev. Mod. Phys. 38 (1966) 143 L. C. Biedenharn in Nuclear spectroscopy, F. Ajzenberg-Selove, Ed. (Academic Press, London, 1960) part B, p. 732 H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306 L. Rosen in Proc. second Int. symp. on polarization phenomena of nucleons (Birkhauser, Basel, 1966) V, 253 P. B. Smith in Nuclear reactions, Vol II, ed. by P. M. Endt and P. B. Smith (North-Holland, Amsterdam, 1962) p. 248 L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711 G. J. McCallum, Phys. Rev. 123 (1961) 568 M. Suffert, P. M. Endt and A. M. Hoogenboom, Physica 25 (1959) 659 A. J. Ferguson in Angular correlation methods in gamma-ray spectroscopy (North-Holland, Amsterdam, 1965) K. W. Dolan, D. K. McDaniels and D. O. Wells, Phys. Rev. 148 (1966) 1151
2~.B
Nuclear Physics A143 (1970) 481 --496; ~ ) North-Holland Publtshlng Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
POLARIZATION AND ANGULAR CORRELATION MEASUREMENTS FOLLOWING THE *°Ar(p, ny)4°K REACTION P. J. TWIN t, W. C. OLSEN and D. M. SHEPPARD
Nuclear Research Centre University of Alberta, Edmonton, Canada tt Received 12 November 1969 Abstract: Gamma-ray angular distributions and polarization correlations have been measured using the reaction 4°At(p, ny)4°K. All the experiments were carried out with bombarding energies close to threshold for the states under investigation. Comparison with compound nuclear predictions has yielded the following spin assignments to states in 4°K; 1.959(2+), 2.047(2-), 2.070(3-), 2.103(1 -), 2.261 (3+), 2.290(1 +), 2.291 (4, (3)), 2.419(2,3), 2.575(2,4) and 2.625(0-). Multiple mixing ratios together with the gamma-ray branching ratios have been determined for all these levels. A short theoretical treatment of the polarization of gamma rays from aligned nuclei using the phase-consistent formalism of Brink and Rose is included. E[
N U C L E A R REACTIONS 4eAr(p, n~'), E = 3-2--5.2 MeV; measured ] tr(E; ET, 0~,),y-polarization. 4OK deduced levels, J, ~, y-branching, ~-mixing. Natural target.
1. Introduction
Recent work by Gerace and Green l) and Flowers and Skouras 2) has focussed attention on the positive parity states around 4°Ca. These states are assumed to be mixtures of 2p-2h, 4p-4h and even 8p-8h configurations. Similar particle-hole states with positive parity should exist in 4°K. We have used the 4°Ar(p, ny)4°K reaction to search for such states and have definitely identified four positive parity states in 4°K at 1.644 MeV (0+), 1.959 MeV (2+), 2.261 MeV (3 +) and 2.290 MeV (1+). Previous work on 4°K, summarized by Endt and van der Leun 3), included the 39K(d, p)4°K experiments of Enge et aL 4). They found the four lowest states at 0, 0.030, 0.800 and 0.891 MeV, were strongly excited via In = 3 transfers, indicating these states were predominately single p-h states of the configuration fld~'t, and hence had negative parity. The states at 2.047, 2.070, 2.103 and 2.625 MeV were also strongly excited but were characterised by In = 1 transfers and therefore were ascribed as negative parity states arising from the p~d~ 1 configuration. All other states they observed below 2.6 MeV had only a very weak excitation cross section in the (d, p) reaction. However, Main et al. 5) reported that the 1.644 MeV state is strongly excited in the 4°Ar(p, n)4°K reaction and measured its lifeti~ae for y-decay to be 1" On leave of absence from the University of Liverpool, England. tt Work supported in part by the Atomic Energy Control Board of Canada. 481
482
P.J. TWINet al.
0.49 Its. They observed the state decayed to both the 0.800 MeV (2-) and 0.030 MeV (3-) states and concluded its spin was probably 0 ÷ ; a 1 ÷ assignment, however, could not be excluded. The 4°Ar(z, t)4°K reaction has recently been investigated by Wesolowski et al. 6) who observed that both the 1.644 and 1.959 MeV states (the lowest states not ascribed to single p-h configurations) were strongly excited. Using a DWBA analysis, they suggest a negative parity assignment for the 1.644 MeV state with J < 2 (inconsistent, however, with the lifetime reported by Main) and positive parity for the 1.959 MeV state with a possible spin of 3. We recently reported 7) measurements of the neutron angular distributions of the low-lying states in 4°K following the 4°Ar(p, n) reaction showing that the 1.644 MeV state has indeed a spin of 0. The report also included a summary of the results of ~-ray angular distribution and polarization experiments identifying three positive parity states in 4°K. The techniques used together with the results for all the lowlying levels in 4°K are described in the present paper.
2. Experimental techniques The proton beam from the University of Alberta Van de Graaff accelerator was collimated to 0.15 cm diameter 15 cm before striking the target gas cell. This cell consisted of two tantalum hemispheres of radii 1.5 cm and l.9 cm with a 7.7 mg ~cm- 2 thick platinum window placed centrally in the hemisphere of larger radius. Thus the beam passed through 0.6 cm of natural argon gas maintained at a pressure of one atmosphere (a 65 keV thick target to 5 MeV protons). For all angular distribution experiments, the system isotropy was checked using the strong 1614 keV v-ray from the decay of the 1.644 MeV (0 ÷) state in 4°K. This ~-ray was observed at all proton energies above 4.3 MeV. 2.1. GAMMA-RAY ANGULAR DISTRIBUTIONS The Ge(Li) detector used in this work had an active volume of 45 cm 3 with a resolution of 3.5 keV for 1332 keV and 4.5 keV for 2615 keV ?-rays. Spectra were recorded at angles of 0, 30, 45, 60 and 90, each angle being repeated at least once. Above a bombarding energy of 4.5 MeV, the intensity of the 1614 keV ?-ray was used as a monitor whilst below this energy the total yield in a NaI crystal was used. The spectra were analysed with a peak-fitting program written by Tepel s) to give the total number of counts in the full energy peaks. The resulting angular distributions were fitted in terms of even-order Legendre polynomials. The a 2 and a 4 coefficients were then compared with the predictions for various spin sequences and multipole mixing ratios, of the compound nuclear statistical model. We used the M A N D ¥ program of Sheldon and Van Patter 9) which uses the Biedenharn 1o) convention for the sign of the mixing ratio c5. This yields the opposite sign for dipole-quadrupole mixing from the Rose and Brink 11) phase-consistent formalism which is used throughout this paper. Neutron and proton transmission coefficients,
'*°ar(P, m')"
483
required by the M A N D Y program, were calculated from the average optical-model parameters of Perey and Buck as tabulated by Rosen 12). The calculated angular distributions were found to be sensitive to the ratio of the l = 1 and I = 0 transmission coefficients for the outgoing neutrons. To minimize this effect, the bombarding energy was kept within 100 keV of threshold so that the magnitude of the I = 1 transmission coefficient was less than 10 ~ that of the l = 0 transmission coefficient. To obtain a quantitative comparison between the compound nuclear predictions and the experimental data, a Z2 analysis using a grid search program was carried out in terms of the initial state's population parameters. These were limited to lie within values calculated at threshold and 200 keV above threshold by the M A N D Y program. This technique enabled angular distributions of all primary T-rays from a state and also 7-T correlations and polarization data to be analysed simultaneously. 2.2. G A M M A - G A M M A A N G U L A R CORRELATIONS
Two 7.6 cm by 7.6 cm NaI crystals, placed 10 cm from the target, were used for the correlation measurements. One was fixed at 90 ° relative to the beam direction whilst the other was moved over an octant such that the angle relative to the beam direction varied from 0 ° to 90 ° and the angle between the planes of each detector and the incident beam varied from 90 ° to 180°. A conventional fast-slow coincidence system was used together with the sum coincidence technique. The total yield of T-rays in the fixed detector was used as a monitor and at least 2 runs were completed at each angle. ANALYSING CRYSTAL INCIDENT PARTICLE
ELECTRIC V E C T O R
TARGET
~
INCIDENT J " GAMMA RAY
SCATTERED GAMMA
f
SCATTERERI \ ANALYSING CRYSTAL
Fig. 1. A schematic representation of the Compton polarimeter indicates the positions of the target, the scattering and the analysing crystals and the electric vector of the ),-ray, together with the angles discussed in the text.
The grid search program (subsect. 2.1) was used to fit the correlation in terms of the population parameters of the initial state. These were again limited to lie within the compound nuclear predictions at threshold and 200 keV above threshold. "Ihe program was based on the tables of Smith 1 3 ) i n which only quadrupole-dipole mixing in the T-rays is considered. ,
484
P.J. TWIN e t al.
2.3. G A M M A - R A Y P O L A R I Z A T I O N M E A S U R E M E N T S
The polarizations of the y-rays were measured by a Compton polarimeter shown schematically in fig. 1. The scattering crystal was a 45 cm a Ge(Li) detector placed 20 cm from the target. The analysing crystals were 7.6 cm diameter by 7.6 cm long NaI detectors placed 12 cm from the scattering crystal and making an angle with respect to the line from the target through the centre of the scattering crystal of = 82 °. One crystal was at ~b = 90 ° and the other at ¢ = 0 ° with respect to the reaction plane. Both crystals were shielded by 15 cm of lead from direct observation of y-rays originating in the target. The complete polarimeter could be moved from 0 = 90 ° to 0 = 0 ° relative to the beam direction. A conventional fast-slow coincidence system was used together with the sum coincidence technique. This resulted in two spectra, as shown in fig. 4, one associated with the scatterer and the 0 ° analyser, and the other associated with the scatterer and the 90 ° analyser. The yields in the full energy peaks, N(O, ¢) were obtained from the spectra. The small difference in the efficiencies of the analysing crystals was corrected for by normalising the yields at 0 = 0 °, where the y-rays are unpolarized. We now present a treatment for the theoretical calculation of the polarization based on the formalism of Rose and Brink 1i). Previous articles by Fagg and Hanna 14) McCallum is), Suffert et al. 16) and Ferguson 17) use various formulations of angular momentum coefficients and polarization definitions. The angular distribution of a 7-ray measured by a detector, insensitive to polarization, from an aligned state of spin J1 with magnetic substrate populations P(M1 ) to a state ']2, M 2 is given by
W(O) = Z P(M1)pk(J1 M1)6"(1 + 62) - 1Rk(LL'J1Sz)QkVk(COS0),
(1)
kM t LL"
where LL' represents the multipolarities (L' _~ L) of the transition, n is 0, 1 and 2 for (LtLI), ( L I L I + I ) and ( L I + I L I + I ) radiation respectively and the Pk and Rk terms are defined and tabulated by Rose and Brink 11). The direction polarization correlation is given by
W(O, T) = •
P(MI)Pk(JI M1)J"(1 +62) - 1Rk(LEJ1 J2)Qk
kMjLL"
x [Pk(cos 0 ) + ( - - ) " COS 2VKk(LE)p2(cos 0)],
(2)
where V is the angle between the reaction plane and the electric vector of the emitted y-ray. The Kk(LL') coefficients are listed by Fagg and Hanna 14) and repeated in table 1 for convenience. If the L' radiation is electric in character then n' = 0 and if L' is magnetic n' = 1. A convenient definition of the theoretical polarization is Pt =
W(O, V = 0°)- W(O, V = 90 °) W(O, V = 0°) + W(O, V = 90°)
which limits Pt to values between + 1 and - 1 with Pt T-ray.
=
(3)
0 indicating an unpolarized
'*°Ar(p,ny)
485
Experimentally the yields in the full energy peaks of the sum coincidence spectra N(O, ~b = 0) and N(O, ~ --- 90) were measured. These correspond in the scattering crystal to experimental Compton scattering intensities for ~, --- 82 ° in the reaction plane and perpendicular to the reaction plane respectively. Hence, neglecting solid angle effects, N(O, c~ = O) = W(O, 7 = 0 ) d a o + W(O, "t = 90)da9o, N(O, q5 = 90) = W(O, 7 = O)da9o+ W(O, y = 90)dao, where dao and do'9o are respectively, the differential cross sections for Compton scattering, at ~, = 82, in and perpendicular to the plane formed by the electric vector and the direction of the incident y-ray. TABLE
l
Numerical values of the coefficientsKk(LL') LL"
k =2
k =4
12 22
-~ ~-
-- -1 2~
Therefore an experimental value for the polarization can be obtained from pg
~---
--
_
1 ,v(o,
= o)-N(o,
= 90)
R N(O, 4) = O) + N(O, ~b = 90)' where, R - da9°-da° do'9o + dtro Hence, using the Klein-Nishina formula R-
sin 2 ~,
ko/k + k/ko - sin2~, '
where k and ko are the wave numbers of the incoming and scattering radiation and ¢, the angle through which the ~,-ray is Compton scattered. R is essentially the polarization efficiency of the polarimeter and can be determined experimentally using y-rays with known multipole mixing ratios. However, Ferguson 17) has suggested a technique for calculating R which takes account of the solid angles subtended by the analysing crystals. At a y-ray energy of 1.5 MeV, the value of R calculated by this method was checked to within 5 ~o using angular distribution and polarization measurements for the 1460 keV 2 + --* 0 + transition in 4°Ar. In the present work, we use these calculated values of R, assuming a nominal 10 ~o error, in determining
P. $. T W I N e t al.
486
polarizations from the experimental N(O, q~) yields. This data was then fitted simultaneously with the angular distributions using the grid search program (subsect. 2.1) which is based on formulas (1) ~ (3) above. The population parameters were restricted to lie within compound nuclear predictions at threshold and 200 keV above threshold. We now show how an estimate of the expected theoretical polarization at 0 = 90 ° can be obtained in terms of the experimentally measured a2 and a4 Legendre coefficients fitted to the angular distribution of the "/-ray. If one considers that only dipole and/or quadrupole de-excitation radiation occurs, then the theoretical polarization measured at 0 = 90 ° taken the following form
E Q2P(M1)P2(J1M1){3[R2(llJ1J2)-F2r~R2(12J1J2)+r~2R2(22J1J2)] MI
-- 8fig2(12J1 J2)} + 1.25 pt =
-I
~ Q4P(M1)p4(JiM1)62R4(E2J1
J2)
MI
-- 2 E P(M1)po(J1M1)(1+62) -
E Q2 P(ML)p2(J1M,)[R2(llJ1J2)
MI
MI
+ 2~R2(12JiJ2) +
62R(22J1J2)]+ 0.75
~
Q4P(M1)P4(JIM1)62R4(22JiJ2)
MI
with the positive sign applicable to MI/E2 mixing and the negative sign to El/M2 mixing. The term
E P(M1)po(J1M,) --- 1
MI
and the terms Ml
Q2P(M~)p2(JIMt)[R2(11J1J2)q-26R2(12J~ J2)+ 52R2(22J1 E Q,~P(M1)p2(J1M~)fi2R4(22J1 Jz)/(1 + fi2),
J2)]/( 1 q- 5 2 ) ,
MI
respectively, are in fact the theoretically predicted a 2 and a4 Legendre polynomial coefficients of the angular distribution of the gamma ray, so that 3a 2 - 8 ~ Q2 P(M1)P2(J1Ml)g2(12Jt J2)6/(1 + 62) + 1.25a4 Pt
=
--
M~
2 - a 2 + 0.75a4 A further simplification is obtained when only pure dipole or quadrupole radiation is present: 3az + magnetic dipole: Pt = - + - - 2- a2 - - electric quadrupole:
Pt = -+"
3a2 + 1.25a4 2 - a 2 + 0.75a4
+ electric -magnetic"
These a2 and a 4 coefficients can then be replaced by their experimental estimates obtained from an analysis of the measured angular distribution. Hence a reasonable
4°At(p, nr)
487
prediction of the theoretical polarization for various spin and parity assignments can be obtained. 3. R ~
The experimentally measured branching ratios and Legendre coefficients together with the spin-parity assignments and associated ~-ray multipole mixing ratios are listed in table 2. The energies of the levels, quoted in the table, have been determined TABLE 2 The results listed are for the states of 4 ° K below 2.65 MeV excited close to the threshold for excitation. The energies of the states have been determined from y-ray measurements in the present work a n d have an error of less than 1.5 keV. The branching ratios have been rounded and have an error 4- 3% Initial state (MeV)
Final state (MeV)
Branching ratio
0.800 1.644
2.291
0.030 0.800 0.030 0.800 0.030 0.800 0.030 0.~0 0.800 0.030 0.000 0.800 0.030 0.030 0.000 1.644 0.800 0.000
100 20 80 85 15 40 30 30 10 55 35 30 70 80 20 65 35 100
2.397
0.030
(50)
2.419
0.000 0.800
(50) 90
2.575
0.030 0.000 0.030
5 5 100
2.625
2.103
100
1.959 2.047
2.070
2.103 2.261 2.290
Legendre coefficients a2 a.
Initial spin
--0.094-0.01
0.004-0.01
20+
0.404-0.01 --0.234-0.01 0.484-0.07 --0.12i0.01 0.16--0.03 --0.6910.10 0.61±0.04 --0.03 4-0.04 --0.124-0.01 0.01 4-0.02 0.414-0.01 --0.10::t=0.02 --0.154-0.03 --0.~4-0.03 0.55-/-0.04
0.00±0.01 0.004-0.02 0.074-0.07 --0.01 i 0 . 0 1 0.00t0.04 0.094-0.12 --0.01±0.05 0.01 4-0.05 0.004-0.01 0.01 +0.02 --0.01 +0.01 0.024-0.03 0.064-0.04 --0.014-0.05 --0.044-0.05
2+ 2-
3-
13÷ 1÷ 4(3)
Final spin 3232323423423340+ 24-
Multipole mixing ratio ¢5 0.004-0.01 M2 E3 0.004-0.02 --0.104-0.04 --0.13+0.09 --0.01 4-0.02 E2 0.204-0.10 --0.274-0.10 0.07 4-0.05 --0.30-t-0.06 E2 --0.024-0.05 0.044-0.06 MI --oo < ~ < --0.3 --0.354-0.25(4)
1.o ±0.3 (3) 30.344-0.03
--0.21 =t:0.03
0.01 4-0.04
0.06-'-0.04
2,3
42-
2,4
343-
0-
1-
--2.0-t-0.6 (2) -0.05--'O.lO(2) -0.354-0.05(3)
--0.08 --0.03(2) --0.06 ~0.02(4) M1
from the measurement of 7-ray energies using the Ge(Li) detector with a 56Co spectrum being taken simultaneously to ensure an accurate calibration. The energies quoted in table 2 have an uncertainty of less than 1.5 keV. The following detailed
488
P.J. TWXNet al.
discussion of the results mentions only a2 Legendre coefficients because none of the angular distributions yielded an appreciable a4 coefficient. 3.1. THE 0.800 MeV STATE
The spin of this state was known to be 2 - and the decay to proceed entirely via a 770 keV y-ray to the 0.030 MeV ( 3 - ) state. However, the E2/M1 multipole mixing d
{o)
1OOOO
100
(b)
100~
X2 100
10 CONFIDENCE LIMIT
:ONFIDENC LIMIT
-90 -45
0
arctan
45
90
lJ
,
-90 -45
,
i
i
0
45
90
arctan
Fig. 2 a) The Zz plot for the 770 keV y-ray angular distribution, measured close to threshold, fitted in terms of the compound nuclear model. b) The 7.2 plot for the angular correlation between the 770 keV and 844 keV y-rays from the 1.644 (0 +) state.
ratio of this y-ray was undetermined. The angular distribution measured at Ep = 3.23 MeV within 40 keV of threshold yielded an a2 of - 0 . 0 9 + 0.01 which could be fitted with two values of 6 as shown by the X2 plot in fig. 2a. To resolve this ambiguity, the Y-7 correlation between the 770 keV and 844 keV -:-rays from the decay of the 1.644 MeV (0 +) state was measured. As the population parameters of the initial state are uniquely determined and the 844 keV 7-ray must be pure quadrupole, the correlation is sensitive only to the mixing ratio of the 770 keV y-ray. The X z plot for the correlation shown in fig. 2b enables the previous ambiguity to be resolved and the 770 keV ),-ray shown to be pure MI. 3.2. THE 1.959 MeV STATE
This state decays 85 Y/ovia a 1159 keV y-ray to the 0.800 MeV ( 2 - ) state and 15 ~o via a 1929 keV y-ray to the 0.030 MeV (3-)state. The angular distributions of both
4°~Lo, m,)
489
~sK 1159 key
~S~~~~~ .),-RAY o') i1
1000
X2
....~-
~J
Z IOK 0U 2 3 4
11[,
,~ 0.1 I[ --H-CONFIDENCEIl LIMIT
1.959 - ~
°'"°°-'-l-r
z
0030"--z-3" "OK 1.OK
~//
y-RAY
I
L0 nrctan ~1159
I
015 cos 2 8
0
Fig. 3. T h e 1.959 M e V state: the Z z plot for the s i m u l t a n e o u s fit o f the a n g u l a r distributions o f t h e 1159 keV a n d 1929 keV 7-rays, m e a s u r e d at a b o m b a r d i n g energy o f 4.44 MeV, 90 keV a b o v e threshold. T h e experimental a n g u l a r distributions together with the best fits for spin a s s i g n m e n t s o f 1, 2, 3 a n d 4 are shown.
(a)
4Kj3!
z
2Kt,0Aa
O
1KF v . ~ . . j ~11,726
I
i~1460 0° ANALYSING II '°A CRYSTAL n59
ol
[ ,,n 3KI Z
(b)
II 1614
1.o
II A
v ~
,0A
Z
~
o
A,.o
t
]J4"OA- 90" ANALYSING II CRYSTAL II 1614
110~3/~
I I It
37Ci
I Kl'-''',--J ~It1,26 ol " ~ CHANNEL
,o~ 2s23
0.5 a=0 12-
, oo
N ,v <~
-0.5
a.
-1.0 9~0°
4'5"
Ir 0
N°.
Fig. 4. T h e 1.959 M e V state a) T h e s u m coincidence spectra at a b o m b a r d i n g energy o f 4.44 MeV, 90 keV above t h r e s h o l d for the 1159 keV y-ray f r o m the 1.959 M e V state, are s h o w n for both 0 ° a n d 90 ° analysing crystals. b) T h e m e a s u r e d polarizations o f t h e 1159 keV y - r a y at angles o f 90 ° a n d 45 ° to the b e a m direction are s h o w n together with the best fits at t h e two angles for spin a s s i g n m e n t s o f 2 a n d 3 when the data was fitted s i m u l t a n e o u s l y with the a n g u l a r distributions o f the I 159 keV a n d 1929 keV y-rays. T h e a n g u l a r distribution data h a d previously indicated the spin o f the state is 2 a n d hence the polarization s h o w s it has positive parity.
490
P. ~. T W I N e t al.
primary ),-rays were measured at a bombarding energy of 4.44 MeV, 90 keV above threshold. They were fitted simultaneously in terms of the 1.959 MeV state's population parameters which were limited to lie within the compound nuclear predictions at threshold and 200 keV above threshold. The resulting Z2 plot shown in fig. 3 clearly indicates that the 1.959 MeV state has a spin of 2 and the 1159 keV 7-ray is pure dipole. The plot shows for each value of 8~59 the minimum X2 obtained when ~1929 is varied from +oo to - o o . Fixing 61159 as zero enables the determination of (~1929 as - 0 . 1 0 + 0 . 0 4 . The y-), correlation between the 770 keV and 1159 keV ),-rays was also measured. However, the correlation yielded no additional information to that obtained from the simple angular distribution of the 1159 keV ),-ray. The explanation is that the only large coefficient in the associated Legendre polynomial expansion of the theoretical ),-)' correlation for this spin sequence is identical with the a 2 Legendre term determined in the simpler experiment. The polarization of the 1159 keV y-ray was measured at angles of 90 ° and 45 ° with the same bombarding energy of 4.44 MeV. The sum coincidence spectra obtained are shown in fig. 4. This data was then fitted simultaneously with the angular distributions of the 1159 and 1929 keV 7-rays. The previous spin assignment o f J = 2 together with the predicted and measured polarizations shown in fig. 4 indicate the 1.959 MeV state must have positive parity. These results confirm the tentative positive parity assignment of Wesolowski 6) bFt show their suggestion of 3 for the spin of the state was incQrrect. 3.3. TI-I~ 2.047, 2.070 AND 2.103 MeV STATES These three states are very strongly excited in the 39K(d, p)4°K reaction with In = 1 transfers and hence were assigned as arising from the p,~d~-1 configuration 4). Four states with spins 0 - , 1-, 2- and 3 - are expected from this configuration and Enge 4) assigned the 2.047 MeV as 3 - , 2.070 MeV as 2 - and 2.103 MeV as l (with the 0 - state at 2.625 MeV) on the basis of excitation cross section. In the present work all three states were strongly excited and their decay modes accurately determined. The 2.047 and 2.070 MeV levels have decay branches to each of 3 lowest states in 4°K whilst the 2.103 MeV state decays only to the 0.030 MeV ( 3 - ) and 0.800 MeV ( 2 - ) states. Angular distributions for all the primary y-rays of the decays of the three levels were obtained at a bombarding energy of 4.55 MeV, between 50 and 100 keV above threshold. The experimental a2 and a4 Legendre coefficients are shown in fig. 5 together with the compound nuclear predictions for final state spins of 2 - , 3- and 4 - . The figure shows that the large a2 coefficients of - 0 . 6 9 40.10 and 0.61 +0.04 for the 1270 and 2040 keV ),-rays indicate the 2.070 MeV state must have a spin of 3. Also the large a2 of 0.48__+0.07 for the 1247 keV ),-ray together with the small a2 of -0.12-t-0.0l for the 2017 keV 7-ray are consistent with a spin of 2 for the 2.047 MeV state, but it is doubtful if a spin assignment of 3 could be completely ruled out. We, therefore, accept the strong evidence that these states do arise
*°Ar(p, n~,)
491
from the pg.d~ ~ configuration and hence assign the 2070 keV state as 3- and the 2.047 1V[eV state as 2-. Furthermore, the anisotropy of the 1303 keV ~. ray indicates the 2.103 MeV state must be the 1- member of this quadruplet. 3.4. T H E 2.261 MeV STATE
This state decays 80 ~ to the 0.030 MeV ( 3 - ) state and 20 % to the 4 - ground state. A possible 1461 keV transition to the 0.800 MeV ( 2 - ) state is completely obscured by the strong 1460 keV v-ray from 4°Ar. Hence a coincidence experiment was carried out with a NaI crystal and the 45 cm 3 Ge(Li). No evidence was found for a 1461 keV 7-ray in coincidence with the 770 keV -?-ray enabling an upper limit of 5 % to be placed on a possible branch to the 0.800 MeV state. PREDICTED 02 A N D 04 LEGENDRE COEFFICIENTS PLOTTED VERSUS B FOR FINAL STATES 2, 3-, 4-
~4 li
---T---J __L_ 4.
-=-.....PURE Q U A D R U P O L E . . . . . . . PURE DIPOLE :.XPERIMEN TAL COEFFICIENTS ~J--- ),-RAYS F R O M 2.047 M e V LEVEL [] .... z-RAYS F R O M 2.070 M e V LEVEL 11ri- -
),-RAYS F R O M 2.103 M e V LEVEL
- 0.8
a4
_J_. J
-0.4
d-----o."-4-- ---o78---*
a4
"~
1-0.8
.a--a--
I
Ir
tI -0.4
1"-0.4
I
°~
J 2-
r
JI
Fig. 5. The a2 and a , Legendre coefficients, predicted by the c o m p o u n d nuclear model, are shown for final state spins o f 2 - , 3 - and 4 - . The multipole mixing ratio 6 varies around the ellipses. The cxperimentally determined coefficients for all the ~,-rays from the 2.047, 2.070 and 2.103 MeV states are plotted.
Angular distributions of the 2231 and 2261 keV 7-rays were measured at a bombarding energy of 4.73 MeV, 70 keV above threshold. When compared with the compound nuclear predictions shown in fig. 5 for final states of 3- and 4 - , the a2 coefficients of 0.41 +_0.01 and - 0 . 1 0 + 0 . 0 2 for the 2231 and 2261 keV 7-rays are only consistent with a spin assignment of 3 for the 2.261 MeV state. The parity of the state was determined by measuring the polarization of the 2231 keV y-ray using the Comp-
492
P.J. Twin
et al.
ton polarimeter which just resolved this ~-ray from the weaker 2261 keV ~-ray. Using the calculated polarimeter efficiency o f 0 . 2 1 + 0 . 0 2 the polarization was determined to be -0.57__+0.30. This data was then fitted together with the 2231 and 2261 keV ~-ray angular distributions. The X2 plot, shown in fig. 6a, clearly establishes the assignment o f 3 + for the 2.261 MeV state. The data further indicates that both the 2231 and 2261 keV 7-rays are predominately E1 transitions. The angular distributions and the polarization measurements together with the best fits for the various spin and parity assignments are shown in fig. 6b. This figure indicates that, in the present case, (b) .~ .~2
2261 keV y-RAY
1o)
1000
0 6K/ , U 1.0
": ::..
•
~6x
44
,\
~.
i
, f 0.5 ¢0S26* 0.0 • 2231 keY ),-RAY
100
L.I
X2
3K
io
~!o o1,
II. o!o •
°3°
1
p .~
"m
ols
COS2
, , . . . . . -I "Or.5 0 ~5 1 2 82231 keV
~
4*
POLARIZATION 0.5| ~ 2 3 " OF 2231 keV )'-RAY O0I"
Fig. 6. The 2.261 MeV state: a) The Z2 plot versus ~ of the 2231 keV),-ray for various spin and parity assignments is shown. The polarization data for the 2231 keV 7'-ray has been fitted together with the angular distribution data for both the 2231 and 2261 keV 7-rays. b) The best fits to the data for the various spin and parity assignments are shown. The fits to the angular distributions are identical for both parity assignments. The results indicate the 2.261 MeV state has a spin of 3 +. the polarization data alone can be equally well fitted by spin assignments o f 2 - , 3 + or 4 - . Hence the spin o f the state must be previously established by the angular distribution data before the polarization measurement enables the parity of the state to be deduced. 3.5. TIlE 2.290 AND 2.291 MeV STATES One state at 2.290 MeV was seen in the 39K(d, p ) 4 ° K reaction. At a b o m b a r d i n g energy o f 4.73 MeV, 40 keV above threshold, we observed possible 7-decay branches to the 1.644 MeV (0+), 0.800 MeV ( 2 - ) , 0.030 MeV ( 3 - ) and g r o u n d ( 4 - ) states, suggesting that the 2.290 MeV state was a doublet. This conclusion was confirmed by
4°At(p, ny)
" 493
j ~
t,O I-Z
~i~
5K
------
-"1 0 U
2*
X 2 • 60.8
2"
X 2,51~4
ca
Z
1÷
X 2 • 0.9
- - - - - - 1-
X 2 " 3.1
5K
0 10
0.5
0
cos 2 0
Fig. 7. The 2.290 MeV state: The angular distribution of the 646 keV 7-ray to the 1.644 MeV (0 +) state is shown together with the best fits for spin assignments 1 +, 1 - , 2 + and 2 - . The state definitely has spin a of I and the fits further indicate that 1 + is the most probable assignment. (b)
|a)
N ~
1000
a2 ,055 ± 0.04
u~ 2500 Z 0 U u_ 2000 0
X 2 100
10 :~ 1500 7
1000
-4;
; arctan
,;
9'0
1.0
05 cos 2 O
0
Fig. 8. The 2.291 MeV state: a) The Z2 plot is shown for various spin assignments to the 2.291 MeV state when the angular distribution of the 7-ray to the 4 - ground state is fitted in terms of the compound nuclear predictions. b) The best fits for spin assignments 3 and 4 are shown together with the experimental angular distribution. These results indicate a spin of 4 for the 2.291 MeV state, although an assignment of 3 cannot be ruled out at the 0.1 ~ confidence limit. the analysis o f the angular distributions, the accurate m e a s u r e m e n t o f the y-ray energies a n d f r o m the o b s e r v a t i o n that the relative intensities o f the four possible
P . J . T W I N et aL
494
decay branches changed at different bombarding energies. From the latter we conclude the following; the 1490 keV ?-ray (to 0.800 MeV state) is associated with the 646 keV ?-ray (to 1.644 MeV state) in the decay of the 2.290 MeV state; the 2291 keV ?-ray is the only predominate decay mode of the 2.291 MeV state; the 2261 keV ?-ray is predominately associated with the 2.261 MeV state and is not a decay from the states at 2.290 and 2.291 MeV to the 0.030 MeV state. The angular distribution of the 646 keV y-ray to the 1.644 MeV (0 ÷) state is shown in fig. 7 together with the best fits for spin assignments of 1 and 2. The population parameters were limited to lie between values calculated at threshold and 200 keV above threshold by the M A N D Y program. These limits are noticeably different for ---1---2.419 2- ' 0.800 / ~ .
/\
40 K
I
-90
I
-45
I
I
I
0
45
90
arctan
Fig. 9. The 2.419 MeV state: The %z plot is shown for various possible spin assignments when the angular distribution ofthe 1619 keVT-rayto the 0.800 MeV (2-) state is compared with the compound nuclear model predictions. the two parity assignments in the case of spin values 1 and 2. Hence fig. 8 shows not only that the 2.290 MeV state must have spin 1 but also that a positive parity assignment is much more likely than a negative parity assignment. The angular distribution of the 2291 keV "t-ray is very anisotropic yielding an az of 0.55__0.04. The Z z plot, shown in fig. 8, for the comparison of the angular distribution with the compound nuclear predictions indicates the 2.291 MeV has a spin of 4, and the multpole mixing ratio of the transition is -0.35___0.25. However, a spin assignment of 3, with a 6 of around unity cannot be completely ruled out at the 0.1 ~o confidence limit.
'*°Ar(p,nT)
495
3.6. THE STATES ABOVE 2.3 MeV The 2.397 MeV state was very weakly excited in the 4°Ar(p, n)4°K reaction. The decay modes observed were approximately equal branches to the 0.030 MeV and the ground state. The 2.419 MeV state predominately decays to the 0.800 MeV ( 2 - ) state with a 1619 keV 7-ray, with weak branches of 5 ~ to both the 0.030 MeV (3 - ) level and the 4 ground state. The angular distribution of the 1619 keV y-ray was measured at a bombarding energy of 4.9 MeV, 70 keV above threshold. The a2 coefficient of 0.34 is consistent with a spin of 2 but as shown in fig. 9, an assignment of 3 cannot be completely ruled out. The 2.575 MeV state is strongly excited and decays entirely to the 0.030 MeV ( 3 - ) state with a 2545 keV ~-ray. The angular distribution of this ?-ray yielded an a2 of - 0 . 2 1 + 0 . 0 3 which is seen from fig. 5 to be consistent with a spin assignment of either 2 or 4. In both cases, the multipole mixing ratio of the ~-ray is close to zero. The 2.625 MeV state was observed to decay entirely to the 2.103 MeV ( 1 - ) state with a 522 keV 7-ray. This adds further weight to its previous identification as the 0 member of the p,ld~- 1 quadruplet due to the strong In = 1 transfer to the state in the 39K(d, p)40 K reaction.
4. Conclusions 4.1. THE NEGATIVE PARITY STATES The only transition between the four lowest fld~-1 states for which we were able to determine c5 is the 770 keV (2- ~ 3 - ) y-ray which is pure M1. This complements previous measurements 3) which show the 30keV ( 3 - - - , 4 - ) y-ray is also M1. We determined the multipole mixing ratios of all 8 transitions between the p~d~. 1 states and the f~d~"~ states. Two of them, the 2047 keV (2- --* 4 - ) and 2073 keV (1- ~ 3 - ) are E2 transitions. The other transitions from the 1- and 2 - states show considerable E2/M1 mixing, see table 2, except for the 2017 keV (2- --, 3 - ) y-ray. All 3 transitions from the 3 - state have large quadrupole components, the smallest being for the 2070 keV (3- ~ 4 - ) y-ray. 4.2. THE POSITIVE PARITY STATES We have established the presence of four positive parity states in 4°K with spins 0 +, 1 +, 2 + and 3 +. They lie at approximately the same excitation energy as the single p-h p~,d~"1 states. Their configuration is probably f~d~-2 or f~d~- 4. However, at present none of the theoretical calculations on p-h states around mass .4 = 40 have been applied to this nucleus. We wish to thank Professor Sheldon for the M A N D Y program and T. P. G. Carola, W. Chung, A. A. Pilt and E. Wong for assistance with the experiment.
496
P.J. TWIN et al.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
W. J. Gerace and A. M. Green, Nucl. Phys. 93 (1967) 110; A123 (1969) 241 B. H. Flowers and L. D. Skouras, Nucl. Phys. A l l 6 (1968) 529 P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 H. A. Enge, E. J. Irwin and D. I-L Weaner, Phys. Rev. 115 (1959) 949 I. G. Main, N. Dawson, D. Kewley, N. Lawley, M. F. Thomas and P. J. Twin, Phys. Lett. 26B (1968) 295 J. J. Wesolowski, L. F. I-Jansen and M.L. Stelts, Phys. Rev. 172 (1968) 1072 P. J. Twin, W. C. Olsen and E. Wong, Phys. Lett. 29B (1969) 570 J. W. Tepel, Nucl. Instr. 40 (1966) 100 E. Sheldon and D. M. Van Patter, Rev. Mod. Phys. 38 (1966) 143 L. C. Biedenharn in Nuclear spectroscopy, F. Ajzenberg-Selove, Ed. (Academic Press, London, 1960) part B, p. 732 H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306 L. Rosen in Proc. second Int. symp. on polarization phenomena of nucleons (Birkhauser, Basel, 1966) V, 253 P. B. Smith in Nuclear reactions, Vol II, ed. by P. M. Endt and P. B. Smith (North-Holland, Amsterdam, 1962) p. 248 L. W. Fagg and S. S. Hanna, Rev. Mod. Phys. 31 (1959) 711 G. J. McCallum, Phys. Rev. 123 (1961) 568 M. Suffert, P. M. Endt and A. M. Hoogenboom, Physica 25 (1959) 659 A. J. Ferguson in Angular correlation methods in gamma-ray spectroscopy (North-Holland, Amsterdam, 1965) K. W. Dolan, D. K. McDaniels and D. O. Wells, Phys. Rev. 148 (1966) 1151