- Email: [email protected]
Study of the 22Ne(d, d)22Ne, 22Ne(d, p)23Ne, 22Ne(d, t)21Ne and 22Ne(p, d)21Ne reactions
Nuclear Physics A95 (1967) 591-607; Not to be reproduced
by photoprint
@ North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
STUDY OF THE 22Ne(d, d?‘Ne, 22Ne(d, p)23Ne, 22Ne(d, 021Ne and 22Ne@, d)21Ne REACTIONS H. F. LUTZ, J. J. WESOLOWSKI, L. F. HANSEN and S. F. ECCLES Lawrence Radiation Laboratory, University of California, Livermore, California t Received
19 December
1966
Abstract: The nuclear reactions *2Ne(d, d)*2Ne, eaNe(d, p)eaNe and azNe(d, t)2eNe induced with 12.1 MeV deuterons and aaNe(p, d)elNe induced with 18.2 MeV protons have been studied by employing a gas target of enriched aeNe and beams accelerated in the Livermore variableenergy cyclotron. The results have been analysed with the nuclear optical model and DWBA calculations. The neutron-transfer reactions have been compared with the predictions of the Nilsson model. The results for the pick-up reactions going to the first three levels of elNe are in good agreement with the predictions of the Nilsson model. The RPC model (rotational and particle motion with Coriolis band mixing) modified for pairing effects was employed in an effort to obtain at least a qualitative explanation of the transitions leading to asNe. There are certain large discrepancies consisting of strong transitions that are predicted theoretically under a wide range of assumptions but not seen experimentally. These results indicate that further detailed calculations are necessary to understand the structure of this nucleus.
E
NUCLEAR REACTIONS. eeNe(d, d), (d, p), (d, t), E = 12.1 MeV and “aNe(p, d), E = 18.2 MeV; measured a(E,, O), a(E,, 6), o(Et, e). Deduced deformation parameter and spectroscopic factors. Enriched target.
1. Introduction The neutron-stripping and pick-up reactions on “Ne leading to states of 23Ne and *‘Ne explore nuclei in the region of light, deformed nuclei ‘) that have received much less attention (probably due to experimental difficulties associated with the use of a gas target) than other odd-mass nuclei with A = 19-31. The reactions reported in this paper are **Ne(d, p)22Ne, *‘Ne(d, p)23Ne and **Ne(d, t)*‘Ne induced with 12.1 MeV deuterons and 22Ne(p, d)*lNe induced with 18.2 MeV protons. Many of the odd-mass nuclei in the A = 19 to A = 31 part of the ld-2s shell have been discussed *) with at least qualitative success in terms of the Nilsson model “). The discussion of *‘Al and *‘Mg by Litherland et al. ‘) is the best known example. Freeman 5, has determined the levels of *‘Ne and *3Ne, except for possible doublets separated by less than 20 keV, by (d, p) reactions on *‘Ne and **Ne. These energy levels were used to analyse the data obtained in the experiment reported in the present paper. Further work on these reactions has been reported recently 6). Howard et al. ‘) have published studies of the *‘Ne(d, py)*lNe reaction and have reported “) work on the **Ne(d, py)* 3Ne reaction. t Work performed under the auspices of the U.S. Atomic Energy Commission. 591
H.
F.
LUTZ
et
at.
The spectroscopic factor S for a one-nucleon transfer reaction has been shown by Satchler ‘) to be simply related to the coefficients (Nlj 1 Owcr) which describe the wave function I C22wct) of a particle in a deformed well in terms of spherical-well, shellmodel wave functions IN&. Erskine lo) has employed (d, p) reactions on heavy deformed nuclei to study the predictions of the RPC model 11) (rotational and particle motion with Coriolis band mixing). It was one ofthe main purposes of the experiment reported in this paper to apply the same techniques to light, deformed nuclei. The situation in light, deformed nuclei is somewhat more difficult to handle than in heavier nuclei because the rotational bands overlap and hole states must be considered in the low-energy part of the spectrum 12). An attempt to include the effects of hole states has been made by using a formulation of the RPC theory modified 13) for. pairing effect r4). This type of calculation is better justified in heavier nuclei where it has been employed, for example, by Berlovich i5) to explain violations in odd-mass deformed nuclei of Alaga et al. 16) intensity rules. Elastic and inelastic angular distributions are analysed in terms of the optical model i7) and extended optical model ‘*’ “). The analysis of the neutron-transfer reactions presented in this paper employs zero-range distorted-wave Born approximation ‘“) (DWBA) with no lower cut-off in the radial integrals. Measured or reasonable optical model parameters were used in the DWBA calculations. Coupled-channels effects 21) in the stripping and pick-up reactions were not included, as these probably are important only when ordinary stripping and pick-up are forbidden. 2. Experimental procedure The data presented in this paper were collected in two runs from deuterons accelerated to 12.1 MeV and protons accelerated to 18.2 MeV. The Livermore variableenergy cyclotron 22) was the source of the accelerated particles. The beam,from the cyclotron was focussed by a‘ pair of quadrupole magnets, bent twice with magnets, and then sent through collimating and antiscattering apertures that defined a beam spot at the centre of the 60.96 cm diam scattering chamber. This beam spot was a rectangle 3.96 mm high and 3.23 mm wide. The beam into the scattering chamber was monitored by means of a Faraday cup and electrometer combination. Neon enriched to 99.7 % “Ne( the rest being 0.19 % “Ne and 0.11 % 20Ne) was contained in a target gas cell 7.62 cm in diam with a 240” continuous window made of 3.3 pm thick Havar foil +. The gas cell was filled ‘to a pressure of 101 mm of Hg for the deuteron runs and to a pressure of 235 mm of Hg for the proton runs. The pressure of the gas was monitored by means of a precision dial manometer and the temperature by a thermocouple in the gas cell. The collimators placed in front of the detection system consisted of a vertical slit 3.23 mm wide placed 14.3 cm from the centre of the scattering chamber, followed by a 3.18 mm diameter hole placed at 21.6 cm. Immediately after this hole, ‘the detector t Obtained
from
Hamilton
Watch
Company,
Metals
Division,
Lancaster,
Pennsylvania,
USA.
=Ne+d
telescope
was placed.
AND
=Ne+p
of a AE (transmission-type
It consisted
593
REACTIONS
surface barrier
diode)
detector and an E (lithium-drifted silicon diode) detector. The AE detector for the deuteron runs was 115 ,um thick operated with a reverse bias of 75 V and for the proton runs 64 jirn thick operated with a reverse bias of 50 V. In both runs the E detector was 2000 pm thick and had a 300 V reverse bias. The particle identification circuit of Goulding et al. 23) was used to select the type of particle to be recorded. The spectra were stored in an 800-channel, pulse-height analyser. The spectra from the (d, p) runs were unfolded by a least-squares numerical computation that fitted each observed spectrum with a sum of Gaussian shapes. This computation was performed on an IBM 7094 electronic computer. The spectra from the other reactions were processed by hand. 3. Experimental
results and distorted waves analysis
In the sections which follow, the experimental data and appropriate theoretical comparisons are presented. For convenience, we first discuss the elastic and inelastic scattering of deuterons by “Ne , then the (p, d) and (d, t) pick-up reactions going to the first three levels of “Ne and, finally, the (d, p) stripping reactions leading to ten known states of 23Ne up to 4 MeV excitation. 3.1. ELASTIC AND INELASTIC
SCATTERING
The differential cross sections for elastic scattering from the ground state of 22Ne(J” = 0’) and the inelastic scattering to the first excited state (J” = 2+) at 1.277 MeV excitation energy are shown in fig. 1. An optical-model program with an automatic parameter search routine, similar to the one described by Maddison 24), was employed to generate the curve shown with the elastic data. The optical model parameters found by the program are given in table 1. These same parameters were used as the optical-model parameters for deuterons in all reaction calculations described in this paper. The complex optical model (for all particles considered in this paper) had a form given by
U(r) = Vj(r)+ iWg(r)+ WDg’(r)+i(hh(r)o’ I+ I/,(r), where
f(r) = [l+exp
(+)]-‘,
R, = r,Af,
g(r) = [ l+exp
(r3)]-‘,
R, = r,Af,
4 exp
r-R, C
g’(r) = [l+exp
(r$)]2’
1
(4)
594
H.
and Vc is the Coulomb
potential
F.
et al.
LUTZ
of a uniformly
charged sphere with a radius given by
R, = r,A+. The above equations calculations.
define spherical
potential
(6)
wells that are required
for the DWBA
The theoretical curve shown with the inelastic data in fig. 1 was calculated from Barrett’s program for the adiabatic approximation rotational optical model 25). In this program, the nuclear shape was taken to be R(e’> = MI
+P2
Y,“(e’)],
and the complex potential (except for the spin-orbit part that was ignored) was expanded in Legendre polynomials coupling the ground and first excited states. The
aIrI”““““““” 0 30 60
90
120 150 180 ‘0 CENTER-OF-MASS
‘II’I 30
‘l’dL9:--
1/60 ANGLE
120
Fig. 1. Angular distributions for elastic and inelastic scattering of 12.1 MeV deuterons from 2aNe. The curve shown with the elastic data is the optical-model fit obtained with the parameters listed in table 1 of the text. These parameters were also used in the DWBA calculations. The curve shown with the inelastic data exciting the Jn = 2+ state of 28Ne was calculated with Barrett’s program for the adiabatic approximation rotational optical model.
same optical parameters that resulted from the search on the elastic data alone were used in this calculation. To be rigorously correct, since this is a coupled-channels calculation one would have to perform a parameter search while attempting to fit both the elastic and inelastic data simultaneously to obtain a modified set of optical parameters that would be valid for the coupled-channels calculation but not the DWBA calculations. The strength of the imaginary part of the complex potential is greater in the ordinary optical model than in the extended optical model because in the former it accounts for all reactions except shape elastic scattering and in the latter for all reactions except the sum of shape elastic and certain inelastic scattering explicitly calculated. Since the main interest of this paper was the study of the singlenucleon transfer reactions, this procedure was not followed. Instead, we have placed a rather large estimated uncertainty of &-20 ‘A on the magnitude of the quadrupole
*‘Ne+d
AND
=‘Ne+p
TABLE
Optical-model
parameters
59.5
REACTIONS
1
for various particles involved in reaction calculations used in this paper
Particle (ML”)
(zl)
$5)
d “)
-101.9
1.289
0.706
P b)
-
58.6
1.25
0.65
t C)
- 120.0
1.33
0.7
(izV)
-29.56 -22.6 -44.0
v, (MeV)
Ref.
(f2)
(f;)
(fz)
1.353
0.594
-8.0
1.3
Present work
1.25
0.47
-7.5
1.25
Perey 2s)
1.33
0.7
1.3
Rook =‘)
“) Optical-model analysis of elastic deuteron data of present work. b) Calculated for Ep = 12 MeV and ,!I = 0.5 with - V = 53.3-0.55(E)+-
- W = 10-t ; “) Optical-model
g
(N-Z)+25
+ ;
(N--Z)+28
j?,
/!I”.
analysis by Rook of lgF(t, t)lsF at Et = 6.8 MeV.
deformation parameter. The shape of the theoretical angular distribution does not vary rapidly with changes in the depths of the potential well. The value for the quadrupole deformation parameter necessary to fit the inelastic data was 0.44. This is to be compared with the value of 0.64 adopted by Stelson and Grodzins 26) from reduced electromagnetic transition probabilities. Stelson and Grodzins used a sharp-edge charge distribution, and as is discussed by Owen and Satchler “), one can deduce a larger value for the quadrupole deformation parameter in this case than when a diffuse-edge nuclear shape is used. However, we place a k20 % uncertainty on our measurement of the quadrupole deformation parameter because of combined uncertainties in the experimental cross sections and nuclear, optical-model parameters. The B(E2) measurement used by Stelson and Grodzins 26) has a probable error of 527 %. The two determinations, therefore, agree to within experimental accuracy without even considering the differences due to the use of a sharp-edge 3.2. PICK-UP
charge distribution REACTIONS
versus a diffuse-edge
LEADING
nuclear
shape.
TO zlNe
The pick-up of a neutron from 22Ne leading to the first three levels of 21Ne was studied with two reactions; the (p, d) reaction (results shown in fig. 2) and the (d, t) reaction (results shown in fig. 3). For each type of reaction the only strong transition is the one leading to the first excited state of 21Ne. Howard et al. ‘) have summarized the spectroscopic data on 21Ne. The first three levels have J” = 2+, 3’ and sf, respectively. The distorted wave theory predictions for the pick-up of a j = 3 neutron using the optical-model parameters in table I are shown in figs. 2 and 3 for transitions going to the 0.349 MeV state of “Ne. The spectroscopic factor deduced by fitting the (p, d) reaction theoretical curve to ex-
596
H.
F.
LUTZ
et
al.
perimental data is 1.86. No attempt was made to deduce an absolute spectroscopic factor from the (d, t) reaction because of uncertainties in the optical-model parameters for the outgoing tritons. With the Nilsson model used, the ground state of “Ne would be described in most simple terms as two neutrons in orbit 7, each with $2” = +jf, outside an inert
Fig. 2. Results for the (p. d) neutron pick-up reaction leading to the first three known states of 21Ne. The proton bombarding energy was 18.2 MeV. Note that the scale for the differential cross section is ten times greater for the reaction to the 0.349 MeV state than for the other two states. The spins and parities of these three states are known to be #+, f+, =$+for the ground, first excited and second excited states, respectively. The Nilsson model predicts a strong transition to the 4 only as is seen in this experiment. The curve is the DWBA prediction for pick-up of a Id+ neutron.
60
0
8’1’1’1’20
40
60
00
‘0 CENTER-OF-MASS
I
I
20
40
I
I
60
I
I
80
ANGLE
Fig. 3. Results for the (d, t) neutron pick-up reaction leading to the first three known states of 21Ne. Note that the scale for the differential cross section is ten times greater for the reaction to the 0.349 MeV state than for the other two states. The curve is the DWBA prediction for the pick-up of a Id+ neutron.
core. The first three levels of “Ne belong to a rotational band built upon an intrinsic state consisting of a single particle in orbit 7. The spectroscopic factor for pick-up of a nucleon from a doubly even, deformed target nucleus is S = 2(Nlj)
Lk~a)~,
(7)
=Ne+d
where the (NZj 1 iha)
AND
=Ne+p
597
REACTIONS
are given by the Nilsson
calculation
3). They are functions
of Nilsson’s deformation parameter q, well-flattening parameter /J and spin-orbit strength parameter K. Harmonic oscillator wave functions are employed in the Nilsson calculations used in this paper. Note that N refers to the number of oscillator quanta in a shell, and in the region of interest the spherical-well orbits Id,, 2s, and Id, all have two oscillator quanta. Nilsson restricts his calculation to a single oscillator shell. The particular orbit of interest (a = 7) has only Id, and Id, spherical-well components with (22% [3+7)’ taking on the value of 1.0 at q = 0 and 0.94 at q = 6. The values that (22% I 3+ 7)’ assumes are 0.0 at u = 0 and 0.06 at q = 6. These considerations predict a strong j = *,I = 2 transition from the ground state of 22Ne to the first excited state of “Ne, and little or no pick-up going to the ground and second excited states of 21Ne. The results of both the (p, d) and (d, t) pick-up reactions are consistent with the above description of the wave functions for the lowlying levels of 21Ne and the ground state of “Ne. 3.3. STRIPPING
REACTIONS
LEADING
TO 23Ne
Ten levels of 23Ne were analysed in the (d, p) stripping reactions. These are all .’Levels up to and including the 3.99 MeV level reported by Freeman 5). The angular
IO2
30 GROUND
‘0
30
60
MeV
STATE
STATE
90
120
150
180
30
60
CENTER-OF-MASS
90
120
150
I80
120
150
180
ANGLE
Fig. 4. The (d, p) stripping results for the reactions leading to the first six known states of 23Ne. The curves are the DWBA predictions calculated with the optical-model parameters listed in table 1 of the text. The spectroscopic factors deduced by normalizing DWBA curves to experimental data are summarized in table 2.
distributions and theoretical curves are displayed in figs. 4 and 5. Two transitions, one going to the 1.70 MeV state and the other to the 2.52 MeV state, exhibit no stripping patterns. The reaction leading to the 1.83 MeV state has been interpreted in the present experiment as a weak j = 3 transfer, although an experiment by Pullen
H.
598
F.
LUTZ
et d.
et al. 6, at a lower bombarding energy (Ed = 7.0 MeV) interpreted this transition as having no stripping pattern. For the low-lying levels reached by I = 2 transitions, it was possible to distinguishj = 3 fromj = 3 transitions by the method given by Lee and Schiffer 30) for I= 2 transitions in the ld-2s shell. The angular distributions displayed in fig. 5 proceeding to the levels of 23Ne at 3.22, 3.43, 3.84 and 3.99 MeV all indicate peaking at O”, making the assignment of l-values less definitive than for the low-lying excited states. The 1 = 2 transitions can
30
**Ne(d p)23Ne 3.43 t&V STATE JZ =2 j=3/2
/I I. LW
90
3.84 L=I
120
150
180
MeV STATE j=3/2
120 150 180 CENTER-OF-MASS
3.99 8=2
MeV
STATE j=3/2
ANGLE
Fig. 5. The (d, p) stripping results for the reactions leading to the seventh through tenth known states of 2SNe. As discussed in the text, there exists other experimental evidence that the 3.22 MeV state has .P = &+. The difference in shape between I = 0 and 1 = 1 DWBA curves is not sufficient at the particular bombarding energy and Q-value to make the assignment unambiguous.
be distinguished from the 1 = 0 and 1 = 1 transitions, but the latter two cause confusion. The assignments made for the 3.22 MeV and 3.84 MeV states were made on rather general expectations of the level sequence expected for negative-parity states and the expected strengths of the transitions populating them. This point will be discussed in detail after the section on the RPC calculation. Pullen et al. 6, have repeated their experiment at Ed = 4.0 MeV and have assigned 1 = 1 to the transition going to the 3.22 MeV level. The work of Howard et al. 8, seems to corroborate this
=Ne+d
assignment.
To indicate
AND
the existence
=Ne+p
REACTIONS
of this contradictory
599
evidence,
we have placed
parentheses around our spin and parity assignment for this level. It should be emphasized that all the spin and parity assignments discussed in this section were made independently of the work of Pullen et al. 6, and Howard et al. *). Also, as will be discussed later, the assignment of I = 1, j = 3 to the 3.22 MeV level creates even more difficulty in interpreting 23Ne within the framework of the Nilsson model. The distorted wave Born approximation code of Macefield 31) was used to generate the theoretical curves for the stripping angular distributions and to extract the spectroscopic factors by comparison with experimental results. These calculations were made on the IBM 7030 (STRETCH) computer at Livermore. The optical model parameters used are given in table 1. The deuteron optical model parameters were found by the search program described in subsect. 3.1. The proton optical model parameters were calculated from the formulae of Perey 28) that account for a large quantity of proton scattering data in the energy region of interest. It was assumed for the calculation of the proton optical model parameters that 23Ne had a quadrupole deformation parameter of 0.5. The cross section can be written
(8) where the factor 1.48 comes from the use of the Hulthtn wave function for the deuteron. Macefield’s code was used to calculate (21,+ 1)/(2ri + 1)0(6),,,. , and comparison with experimental data gave the spectroscopic factor S. Two sets of stripping calculations were performed which differed from one another only in the manner of treating the bound-state neutron wave function. In the first set, the radius parameter was varied while the depth of the well was kept fixed at 45 MeV. In the second set of calculations, the eigenfunction was generated in a real potential well with the Woods-Saxon shape whose size was determined by R = 1.25 A’ and whose depth was varied until the correct binding energy was obtained. Pinkston and Satchler 32) h ave stressed the importance of using a single-particle wave function with the correct asymptotic form and have stated that both the abovementioned procedures are equally correct. In this paper the two procedures are used to ascertain the degree of uncertainty in the determination of the spectroscopic factors due to the uncertainty in describing the correct wave functions for the captured neutron 3 “). The spectroscopic factors deduced by DWBA analysis of the 22Ne(d, p)23Ne stripping reactions are presented in table 2. For the ground state of 23Ne, VN = -45 MeV when rn = 1.25 fm. This fortuitous situation resulted in no difference between the two procedures for handling the bound-state neutron wave function in the case of the transition to the ground state. For the other transitions, the larger value of the spectroscopic factor was usually deduced for the case when rN = 1.25 fm and the potential depth VN varied. The single exception is the level at 2.31 MeV.
600
H. F. LUTZ et d.
Note the very large error due to uncertainty in treating the bound-state wave function for the transition going to the negative-parity state at 3.84 MeV excitation energy. This transition selects out the 2 pq component of the wave function for the 3.84 MeV level. Why this level should occur so low in the spectrum is discussed in the following section. The uncertainty in treating this level which is displaced in energy from its normal spherical shell-model location is the same type of uncertainty encountered by Sherr et al. 34) in th eir study of (p, d) reactions to isobaric analogue levels. The problem has been discussed in detail by Pinkston and Satchler 32). TABLE
Spectroscopic
QWeV)
.WMeV)
2
factors deduced for eeNe(d, p)2SNe reactions Nlj
s
A(%) -
0.0
2.96
0.23 0.23
1.02
1.94
1.83
1.13
2.31
0.65
0.37 0.40 0.020 0.026 0.077 0.064
3.22
-0.26
3.43
-0.47
3.84
-0.88
3.99
-1.03
Remarks for this level only, V, = -45 MeV when rN = 1.25 fm
5 15
0.64 0.81 0.35 0.40 0.04 0.24
9
for this level only S, obtained with V, = -45 MeV
14
other experiments have assigned Jn = t- to this level
9 71
negative-parity state belonging to band based on Nilsson orbit 14
5
0.30 0.33
As explained in the text, two sets of calculations were performed which differed in the manner of treating the bound-state neutron wave function. In this table, the first entry in the column for S is the case when r~ varied and V, was held constant; the second entry is for r, constant and V, varied. The following definitions are used: A
where S< (S,) calculations.
=
@-‘<)
-
x
‘00
S
is the larger (smaller) value of the spectroscopic
factor resulting
from these
4. RPC calcula@on for 23Ne energy levels and spectroscopic factors The expression derived by Satchler 9, for the spectroscopic factor for stripping of a nucleon leading to a nucleus described as an odd-nucleon strongly coupled to an axially-symmetric deformed core is
22Ne+d
where g is initial and assumed to describing
AND
=Ne+p
601
REACTIONS
42 for a doubly even target nucleus and (Q2 I Q1> the overlap of the final vibrational parts of the total wave function. This overlap is usually be unity for transitions to low-lying levels. The (NZj 1 QOCY)are coefficients the deformed-well eigenfunctions in terms of the spherical-well, shell-
model wave functions IQC0a) = c (NZj~GBa>~NZj),
(10)
Nlj
where the summation is shell, N is 2. The symbol the symmetry axis of the The coefficients obey the
usually restricted to a single oscillator shell. For the ld-2s Q is the projection of the total spin of the odd nucleon onto potential well, w the parity and a the Nilsson orbit number. sum rules &(NZjlQwa>’
The Hamiltonian that rotation-particle coupling
= 1,
(11)
G~~(NZjlQwa)z = 2j+l.
(121
describes is
H = Hintr+
the rotational
$(Z’-21;jb)+
motion
$(l;jl,+ri,
of the nucleus
j;),
and the
(13)
Hi,,, is the part of the Hamiltonian describing the intrinsic states and has been solved by Nilsson 3, with the input parameters (q, p, JC). The primes denote the fact that the spherical components of I and j are referred to body-centred coordinates. It is convenient to allow A, = h2/29= where the subscript a refers to a particular intrinsic state. Neglecting the last term of eq. (13) gives the basic set of wave functions IIMKa)
=
[
1
f{D~,lKa)+(-l)r-‘D~-,l-Ka)~,
$$
where the DLK are rotational wave functions, and a is necessary to label different intrinsic states with the same value of K. The [Ka) are Nilsson wave functions defined as in eq. (10). The last term in eq. (13) is responsible for coupling states with the same value of I with AK = f 1 or AK = 0 if, and only if, K = K’ = +. The diagonal elements of the Hamiltonian are E(NlKa)
= E~+A,{[I(I+I)+a(-1)‘+~(I+~)&3]-[K(K+1)-a6K~]},
(15)
where E:: = E,, + A,[K - KZ - u&~] ; and the decoupling
parameter
(16)
is
a = - c (NZjlS2coa)2(j+$)(-l)j+*.
(17)
II.F.
602
The off-diagonal
elements
LUTZ
of the Hamiltonian
I acrra = (IMK’a’J2(1;
.?t al.
are
jl_, +I’_, j;lzMKa)
= -JU(Z+K,)(Z-K>+l) X[1-6K’K+(-1)r-~~Kf8K)]a.‘.,
(18)
where K, is the greater of K’, K, and A is [+(A, + A,)]; a,,a = 7 (N2jl~oa’>
matrix
is diagonalized
(19)
to give the wave functions
IZM) = 1 CJZMKa) Ka
and the eigenvalues
for each mixed state with spin I. With band mixing,
expression
(9) becomes
(21) The effects of pairing
are included
JSlj =
by writing
[A]’ f
E G, U,(NMoa),
(22)
where U,” is the probability that a particular state is filled 14). The off-diagonal elements are modified by the effects of pairing by multiplying each off-diagonal element by (U,. U,+ V,. V,), where 17, and V, are related by
u,z+v,z = 1.
(23)
described A computer code developed by Erskine 35) performed the calculations above. The input data for the calculation is given in table 3. The RPC calculation (results summarized in table 4) included levels belonging to rotational bands built on five intrinsic states that could be expected, from inspection of the Nilsson level diagram, to make contributions to the positive-parity levels of 23Ne considered in this paper. Efforts were made to minimize the number of parameters varied. The A, associated with the amounts of inertia are similar in value to others used in calculations in this region of the nuclear mass table. The parameters U, were chosen to produce the effects of pairing in at least a qualitative fashion. The values for U, are felt to be reasonable, but they are not the result of any further calculation. The outstanding differences that resulted between experiment and the calculation were not dependent on the choice of U,. This was ascertained by performing the calculation without considering hole states. Various calculations were made for q increasing in steps of 0.5. The value 3.0 was the largest value that could be used with harmonic oscillator wave functions since, for larger values of q, the Nilsson orbit 9 falls at a
“Ne+
d AND
“Ne +
603
p REACTIONS
TABLE 3 Input data for RPC calculation with pairing corrections Orbit number a
Asymptotic quantum numbers Nn,A Li
Energy from Nilsson level (MeV)
202 +
5
Base energy (MeV)
(I&)
uz,
0.
0.
0.3
0.5
9
211:
0.39
0.66
0.15
0.8
7
211%
2.26
1.46
0.15
0.2
11
200 &
2.94
2.82
0.15
1.0
8
202 #
3.98
3.48
0.15
1.0
Nilsson wave functions were generated with 17= 3, ,U = 0.2 and K = 0.06. The decoupling parameter for orbit 9 is -0.2445 and for orbit 11 is -0.2817.
TABLE
4
Results of RPC calculation for positive-parity levels showing spins, energy levels with Coriolis band mixing considered, the strengths of transitions with and without band mixing and the wave functions as defined by eq. (19) of the text CKca
@J-t 1 )S
QJ+
CK,
CK,
cK,
CK,
1)s
K=@
K=:
K=f
K=:
a=5
a=9
a=7
a=
K=$
11
a=8
(lZV)
mixed
unmixed
1.07 3.27
0.67 0.89
0.52 1.04
-0.995 0.097
1.25 1.93 3.56 4.07
1.42 0.24 1.43 0.58
0.83 0.01 0.90 0.94
-0.942 0.227 0.246 -0.034
-0.250 -0.966 -0.069 -0.014
-0.180 0.115 -0.859 -0.465
-0.135 0.054 -0.443 0.885
0 2.05 3.13 4.16 5.29
1.44 0.28 0 0 0.03
1.0 0.25 0.39 0.06 0.06
0.924 0.162 0.342 0.050 -0.021
-0.060 0.922 -0.266 0.039 0.273
-0.316 0.232 0.893 0.089 -0.007
0.016 -0.223 -0.002 0.724 -0.653
-0.032 0.143 -0.124 0.681 -0.706
1.33 2.62 4.12 5.55 6.17
0 0 0 0 0
0 0 0 0 0
-0.796 0.200 0.526 -0.218 0.049
0.168 0.879 0.103 0.431 -0.054
0.577 -0.023 0.736 -0.353 0.002
0.021 0.342 -0.337 -0.693 -0.536
0.069 0.264 -0.241 -0.403 -0.840
2.84 4.37 7.37 8.40 6.22
0 0 0 0 0
0 0 0 0 0
0.644 0.287 0.686 -0.118 -0.136
-0.272 0.883 -0.035 0.375 0.072
-0.704 -0.139 -0.673 -0.091 -0.155
0.092 -0.286 0.273 0.640 0.652
-0.085 0.194 0.030 -0.654 0.726
0.097 0.995
604
H.
F.
LUTZ
et
cd.
lower energy than does orbit 5. This would give a ground state J" = +' instead of 5 + as is known from experiment. The value (0.2) taken for p is the one suggested by &shop 36). The value of K determines the d,-d, splitting. It was found that IC = 0.06
-
aii
;-r -
1
‘\-’
-\
I I-
-’ ---
i -
1 0t -\
'K-5/2 a-5
-\= ;;2 a.9
_I’ -.-
-
--K*l12 cr.11
K=3/2 a.9
K=3/2 a-7
Fig. 6. The effect of the Coriolis band mixing on the position of the positive parity levels considered in the RPC described in the text and summarized in tables 3 and 4. For each band, the level sequence without Coriolis band mixing is shown on the left-hand side, and then the position of the levels after the Coriolis interaction is considered is shown on the right-hand side. The ground state is depressed by several hundred keV. This is true only if the hole in orbit 7 is included in the calculation. THEORY
0
EXPERIMENT
I (2J +I)S
Fig. 7. Comparison
2
0
I (2J+I)S
2
of the results of the DWBA analysis and RPC calculation for the positive-parity levels of 2SNe.
would place orbit 8 near 4 MeV excitation energy, as indicated by the stripping results. The values of the decoupling parameters for orbits 9 and 11 were calculated from the Nilsson wave functions generated for the intrinsic states with the chosen set of parameters (q, p, K). They were not treated as parameters to be varied. The base
=Ne+d
AND
-=Ne+p
605
REACTIONS
energies for the rotational bands were varied slightly from the Nilsson level values. The effects of the Coriolis band mixing on the position of the positive parity levels is shown in fig. 6. For each band, the level sequence without Coriolis band mixing is shown on the left-hand side, and then the position of the levels after the Coriolis interaction is considered is shown on the right-hand side. Note that the ground state is depressed by several hundred keV. This is due largely to the Coriolis interaction between ground state (1 = 4, K = $) and the second member of the band built on orbit 7 (I = 4, K = 3). Orbit 7 must be included in the calculation to depress the ground state and give it a K = $ component. The results of the RPC calculation for the positive parity levels of 23Ne are summarized in table 4. The results of the DWBA analysis of the experimental data and the RPC calculation are compared in fig. 7. The RPC calculation predicts that we should see eight transitions with (2J+ 1)s greater than 0.1 in the energy region scanned by this experiment proceeding to positive-parity states. The data reveal at most seven. (If the 3.22 MeV level is negative parity, then only six are seen.) The TABLE
Results
of RPC calculation
for negative-parity
5
transition
belonging
(2J+ 11.9 mixed
4.87 3.62 5.84 3.75
0.07 0.52 0.18 3.39
to band
built on Nilsson
orbit 14
(2J+ 1)s unmixed
0.06 0.48 0.04 1.41
The Nilsson parameters were 7 = 4.5, ,u = 0.35 and K = 0.06. The parameter A, was taken to be 0.150 MeV. The base energies for the bands were taken from the Nilsson calculation. The decoupling parameter was -3.205.
missing one is a strongj = 3, 1 = 2 transition to a state above the first J” = 4 level. Some RPC calculations were performed by using only four intrinsic states and omitting the hole-state in Nilsson orbit 7. The weak j = 3 transition to a level around 2 MeV excitation energy was deleted in this fashion, but the predictions of this truncated calculation for the other seven transitions agreed qualitatively with the predictions of the full calculation. Especially evident was the prediction of a strong transition, (2J+ 1)s = 2, going to a 3’ level near the first rlf level. In fact, it was found that this transition is predicted whether one considers orbit 7 or not and whether or not Coriolis band mixing is included. One possible explanation is suggested by examination of fig. 6. The 4 and f members of the K = 3 band built or orbit 9 are pushed together by the Coriolis band mixing. It is possible that they may actually be so close in energy that they cannot be resolved experimentally. The DWBA fit to the angular distribution for protons going to the J” = &+ state at 1.02 MeV could be improved by adding a j = 3, 1 = 2 angular
H.
606
F.
LUTZ
et
al.
distribution of about the magnitude suggested by the RPC calculations. (A strength (2J+ 1)s of 1.4 implies a peak cross section of about 7 mb/sr.) The comparison of theory and experiment suggests that the two levels we see that do not exhibit the usual stripping pattern are the 3’ members of the ground-state band and the band built on orbit 9. An RPC calculation was also performed for the negative parity states of the N = 3, lf-2p shell. All ten intrinsic Nilsson states for the N = 3 shell were included in the computation made for r] = 4.5, p = 0.35 and rc = 0.06. The parameters A, were all taken to be 0.150 MeV and the positions of the band heads taken from the Nilsson calculation. With the above mentioned values for the Nilsson parameters (q, ,u, rc), the decoupling of orbit 14 was calculated to be -3.205. This is large enough to produce inversion of the ordering of the levels for the band. The results are summarized in table 5. One expects to see a level sequence 3-, I-, &- and 8- with strong transitions going to the $- and 3- levels. If the 3.22 MeV state has J” = -&-then two 1 = 1 transitions are seen in the region studied. The level sequence for a decoupling parameter in the range - 4 to - 3 (for orbit 14 this corresponds to q varying from 0 to 6) is +,$,t, 3 implying that an 1 = 3 transition should occur between the two 1 = 1 transitions. Furthermore, if we fit the 3.22 MeV angular distribution with an 1 = 1,j = 3 DWBA curve, the value deduced for (2J+ 1)s is about 1.7. This is about three times larger than any I = 1 transition listed in table 5. 5. Summary The neutron pick-up and stripping reactions studied in this paper have investigated the low-lying levels of ‘rNe and 23Ne. The transitions to “Ne populate the ground state rotational band and agree well with the predictions of the Nilsson model. The RPC calculations for the structure of 23Ne, as employed in the present paper, are not expected to give detailed agreement with experimental data; but the discrepancies revealed by the stripping reactions leading to levels of 23Ne disclose certain major discrepancies between the experimental data and the theoretical results. These discrepancies consist of strong transitions predicted theoretically but not seen experimentally, and they persist whether or not one considers the hole-state in orbit 7 and whether or not Coriolis band mixing is included. The authors would like to express their sincere thanks to Dr. J. R. Erskine for the use of his RPC computer code and for several useful communications, to Dr. B. E. F. Macefield for his DWBA code, to Dr. R. Barrett for his adiabatic -approximation scattering code and Dr. D. J. Pullen and Dr. A. J. Howard for receipt of data prior to their publication. Note added in prooJ
bound-state
Dr. Satchler has informed the authors that, in treating the wave function, Pinkston and Satchler 32) suggested that varying the
“Ne+
d AND
B2Ne+p
REACTIONS
607
radius of the potential might be more physically correct than varying the depth and that both procedures give the correct asymptotic tail. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 3 1) 32) 33) 34) 35) 36)
G. Rakavy, Nuclear Physics 4 (1957) 375 R. K. Sheline and R. A. Harlan, Nuclear Physics 29 (1962) 177 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) A. E. Litherland et al., Can. J. Phys. 36 (1958) 378 J. M. Freeman, Phys. Rev. 120 (1960) 1436 D. J. Pullen, A. Sperduto and E. Kashy, Bull. Am. Phys. Sot. 10 (1965) 38 A. J. Howard, D. A. Bromley and E. K. Warburton, Phys. Rev. 137 (1965) B32 A. J. Howard et al., Bull. Am. Phys. Sot. 11 (1966) 406 G. R. Satchler, Ann. of Phys. 3 (1958) 275 J. R. Erskine, Phys. Rev. 138 (1965) B66 A. K. Kerman, in Nuclear reactions, Vol. I, ed. by P. M. Endt and M. Demeur (North-Holland Publ. Co., Amsterdam, 1959) chapt. X H. Lancman, A. Jasinski, J. Kownacki and J. Ludziejewski, Nuclear Physics 69 (1965) 384 D. Kurath, private communication S. Yoshida, Phys. Rev. 123 (1961) 2122 E. Ye. Berlovich, Phys. Lett. 13 (1964) 161 G. Alaga, K. Alder, A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 9 (1955) P. E. Hodgson, The optical model of elastic scattering (Oxford at the Clarendon Press, 1963) D. M. Brink, Proc. Phys. Sot. A68 (1955) 994 S. Yoshida, Progr. Theor. Phys. 19 (1958) 169 W. Tobocman, Theory of direct nuclear reactions, (Oxford University Press, 1961) S. K. Penny and G. R. Satchler, Nuclear Physics 53 (1964) 145 H. P. Hernandez, J. M. Peterson, B. H. Smith and C. J. Taylor, Nucl. Instr. 9 (1960) 287 F. S. Goulding, D. A. Landis, J. Cerny and R. H. Pehl, Nucl. Instr. 31 (1964) 1 R. N. Maddison, Proc. Phys. Sot. 79 (1962) 264 R. C. Barrett, Nuclear Physics 51 (1964) 27 P. H. Stelson and L. Grodzins, Nucl. Data Al (1965) 21 L. W. Owen and G. R. Satchler, Nuclear Physics 51 (1964) 155 F. G. Perey, Phys. Rev. 131 (1963) 745 J. R. Rook, Nuclear Physics 61 (1965) 219 L. L. Lee, Jr. and J. P. Schiffer, Phys. Rev. 136 (1964) B405 B. E. F. Macefied, private communication W. T. Pinkston and G. R. Satchler, Nuclear Physics 72 (1965) 641 J. R. Erskine et al., Phys. Rev. Lett. 14 (1965) 915 R. Sherr, E. Rost and M. E. Rickey, Phys. Lett. 12 (1964) 420 J. R. Erskine, private communication G. R. Bishop, Nuclear Physics 14 (1959) 376
by photoprint
@ North-Holland
Publishing
Co., Amsterdam
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STUDY OF THE 22Ne(d, d?‘Ne, 22Ne(d, p)23Ne, 22Ne(d, 021Ne and 22Ne@, d)21Ne REACTIONS H. F. LUTZ, J. J. WESOLOWSKI, L. F. HANSEN and S. F. ECCLES Lawrence Radiation Laboratory, University of California, Livermore, California t Received
19 December
1966
Abstract: The nuclear reactions *2Ne(d, d)*2Ne, eaNe(d, p)eaNe and azNe(d, t)2eNe induced with 12.1 MeV deuterons and aaNe(p, d)elNe induced with 18.2 MeV protons have been studied by employing a gas target of enriched aeNe and beams accelerated in the Livermore variableenergy cyclotron. The results have been analysed with the nuclear optical model and DWBA calculations. The neutron-transfer reactions have been compared with the predictions of the Nilsson model. The results for the pick-up reactions going to the first three levels of elNe are in good agreement with the predictions of the Nilsson model. The RPC model (rotational and particle motion with Coriolis band mixing) modified for pairing effects was employed in an effort to obtain at least a qualitative explanation of the transitions leading to asNe. There are certain large discrepancies consisting of strong transitions that are predicted theoretically under a wide range of assumptions but not seen experimentally. These results indicate that further detailed calculations are necessary to understand the structure of this nucleus.
E
NUCLEAR REACTIONS. eeNe(d, d), (d, p), (d, t), E = 12.1 MeV and “aNe(p, d), E = 18.2 MeV; measured a(E,, O), a(E,, 6), o(Et, e). Deduced deformation parameter and spectroscopic factors. Enriched target.
1. Introduction The neutron-stripping and pick-up reactions on “Ne leading to states of 23Ne and *‘Ne explore nuclei in the region of light, deformed nuclei ‘) that have received much less attention (probably due to experimental difficulties associated with the use of a gas target) than other odd-mass nuclei with A = 19-31. The reactions reported in this paper are **Ne(d, p)22Ne, *‘Ne(d, p)23Ne and **Ne(d, t)*‘Ne induced with 12.1 MeV deuterons and 22Ne(p, d)*lNe induced with 18.2 MeV protons. Many of the odd-mass nuclei in the A = 19 to A = 31 part of the ld-2s shell have been discussed *) with at least qualitative success in terms of the Nilsson model “). The discussion of *‘Al and *‘Mg by Litherland et al. ‘) is the best known example. Freeman 5, has determined the levels of *‘Ne and *3Ne, except for possible doublets separated by less than 20 keV, by (d, p) reactions on *‘Ne and **Ne. These energy levels were used to analyse the data obtained in the experiment reported in the present paper. Further work on these reactions has been reported recently 6). Howard et al. ‘) have published studies of the *‘Ne(d, py)*lNe reaction and have reported “) work on the **Ne(d, py)* 3Ne reaction. t Work performed under the auspices of the U.S. Atomic Energy Commission. 591
H.
F.
LUTZ
et
at.
The spectroscopic factor S for a one-nucleon transfer reaction has been shown by Satchler ‘) to be simply related to the coefficients (Nlj 1 Owcr) which describe the wave function I C22wct) of a particle in a deformed well in terms of spherical-well, shellmodel wave functions IN&. Erskine lo) has employed (d, p) reactions on heavy deformed nuclei to study the predictions of the RPC model 11) (rotational and particle motion with Coriolis band mixing). It was one ofthe main purposes of the experiment reported in this paper to apply the same techniques to light, deformed nuclei. The situation in light, deformed nuclei is somewhat more difficult to handle than in heavier nuclei because the rotational bands overlap and hole states must be considered in the low-energy part of the spectrum 12). An attempt to include the effects of hole states has been made by using a formulation of the RPC theory modified 13) for. pairing effect r4). This type of calculation is better justified in heavier nuclei where it has been employed, for example, by Berlovich i5) to explain violations in odd-mass deformed nuclei of Alaga et al. 16) intensity rules. Elastic and inelastic angular distributions are analysed in terms of the optical model i7) and extended optical model ‘*’ “). The analysis of the neutron-transfer reactions presented in this paper employs zero-range distorted-wave Born approximation ‘“) (DWBA) with no lower cut-off in the radial integrals. Measured or reasonable optical model parameters were used in the DWBA calculations. Coupled-channels effects 21) in the stripping and pick-up reactions were not included, as these probably are important only when ordinary stripping and pick-up are forbidden. 2. Experimental procedure The data presented in this paper were collected in two runs from deuterons accelerated to 12.1 MeV and protons accelerated to 18.2 MeV. The Livermore variableenergy cyclotron 22) was the source of the accelerated particles. The beam,from the cyclotron was focussed by a‘ pair of quadrupole magnets, bent twice with magnets, and then sent through collimating and antiscattering apertures that defined a beam spot at the centre of the 60.96 cm diam scattering chamber. This beam spot was a rectangle 3.96 mm high and 3.23 mm wide. The beam into the scattering chamber was monitored by means of a Faraday cup and electrometer combination. Neon enriched to 99.7 % “Ne( the rest being 0.19 % “Ne and 0.11 % 20Ne) was contained in a target gas cell 7.62 cm in diam with a 240” continuous window made of 3.3 pm thick Havar foil +. The gas cell was filled ‘to a pressure of 101 mm of Hg for the deuteron runs and to a pressure of 235 mm of Hg for the proton runs. The pressure of the gas was monitored by means of a precision dial manometer and the temperature by a thermocouple in the gas cell. The collimators placed in front of the detection system consisted of a vertical slit 3.23 mm wide placed 14.3 cm from the centre of the scattering chamber, followed by a 3.18 mm diameter hole placed at 21.6 cm. Immediately after this hole, ‘the detector t Obtained
from
Hamilton
Watch
Company,
Metals
Division,
Lancaster,
Pennsylvania,
USA.
=Ne+d
telescope
was placed.
AND
=Ne+p
of a AE (transmission-type
It consisted
593
REACTIONS
surface barrier
diode)
detector and an E (lithium-drifted silicon diode) detector. The AE detector for the deuteron runs was 115 ,um thick operated with a reverse bias of 75 V and for the proton runs 64 jirn thick operated with a reverse bias of 50 V. In both runs the E detector was 2000 pm thick and had a 300 V reverse bias. The particle identification circuit of Goulding et al. 23) was used to select the type of particle to be recorded. The spectra were stored in an 800-channel, pulse-height analyser. The spectra from the (d, p) runs were unfolded by a least-squares numerical computation that fitted each observed spectrum with a sum of Gaussian shapes. This computation was performed on an IBM 7094 electronic computer. The spectra from the other reactions were processed by hand. 3. Experimental
results and distorted waves analysis
In the sections which follow, the experimental data and appropriate theoretical comparisons are presented. For convenience, we first discuss the elastic and inelastic scattering of deuterons by “Ne , then the (p, d) and (d, t) pick-up reactions going to the first three levels of “Ne and, finally, the (d, p) stripping reactions leading to ten known states of 23Ne up to 4 MeV excitation. 3.1. ELASTIC AND INELASTIC
SCATTERING
The differential cross sections for elastic scattering from the ground state of 22Ne(J” = 0’) and the inelastic scattering to the first excited state (J” = 2+) at 1.277 MeV excitation energy are shown in fig. 1. An optical-model program with an automatic parameter search routine, similar to the one described by Maddison 24), was employed to generate the curve shown with the elastic data. The optical model parameters found by the program are given in table 1. These same parameters were used as the optical-model parameters for deuterons in all reaction calculations described in this paper. The complex optical model (for all particles considered in this paper) had a form given by
U(r) = Vj(r)+ iWg(r)+ WDg’(r)+i(hh(r)o’ I+ I/,(r), where
f(r) = [l+exp
(+)]-‘,
R, = r,Af,
g(r) = [ l+exp
(r3)]-‘,
R, = r,Af,
4 exp
r-R, C
g’(r) = [l+exp
(r$)]2’
1
(4)
594
H.
and Vc is the Coulomb
potential
F.
et al.
LUTZ
of a uniformly
charged sphere with a radius given by
R, = r,A+. The above equations calculations.
define spherical
potential
(6)
wells that are required
for the DWBA
The theoretical curve shown with the inelastic data in fig. 1 was calculated from Barrett’s program for the adiabatic approximation rotational optical model 25). In this program, the nuclear shape was taken to be R(e’> = MI
+P2
Y,“(e’)],
and the complex potential (except for the spin-orbit part that was ignored) was expanded in Legendre polynomials coupling the ground and first excited states. The
aIrI”““““““” 0 30 60
90
120 150 180 ‘0 CENTER-OF-MASS
‘II’I 30
‘l’dL9:--
1/60 ANGLE
120
Fig. 1. Angular distributions for elastic and inelastic scattering of 12.1 MeV deuterons from 2aNe. The curve shown with the elastic data is the optical-model fit obtained with the parameters listed in table 1 of the text. These parameters were also used in the DWBA calculations. The curve shown with the inelastic data exciting the Jn = 2+ state of 28Ne was calculated with Barrett’s program for the adiabatic approximation rotational optical model.
same optical parameters that resulted from the search on the elastic data alone were used in this calculation. To be rigorously correct, since this is a coupled-channels calculation one would have to perform a parameter search while attempting to fit both the elastic and inelastic data simultaneously to obtain a modified set of optical parameters that would be valid for the coupled-channels calculation but not the DWBA calculations. The strength of the imaginary part of the complex potential is greater in the ordinary optical model than in the extended optical model because in the former it accounts for all reactions except shape elastic scattering and in the latter for all reactions except the sum of shape elastic and certain inelastic scattering explicitly calculated. Since the main interest of this paper was the study of the singlenucleon transfer reactions, this procedure was not followed. Instead, we have placed a rather large estimated uncertainty of &-20 ‘A on the magnitude of the quadrupole
*‘Ne+d
AND
=‘Ne+p
TABLE
Optical-model
parameters
59.5
REACTIONS
1
for various particles involved in reaction calculations used in this paper
Particle (ML”)
(zl)
$5)
d “)
-101.9
1.289
0.706
P b)
-
58.6
1.25
0.65
t C)
- 120.0
1.33
0.7
(izV)
-29.56 -22.6 -44.0
v, (MeV)
Ref.
(f2)
(f;)
(fz)
1.353
0.594
-8.0
1.3
Present work
1.25
0.47
-7.5
1.25
Perey 2s)
1.33
0.7
1.3
Rook =‘)
“) Optical-model analysis of elastic deuteron data of present work. b) Calculated for Ep = 12 MeV and ,!I = 0.5 with - V = 53.3-0.55(E)+-
- W = 10-t ; “) Optical-model
g
(N-Z)+25
+ ;
(N--Z)+28
j?,
/!I”.
analysis by Rook of lgF(t, t)lsF at Et = 6.8 MeV.
deformation parameter. The shape of the theoretical angular distribution does not vary rapidly with changes in the depths of the potential well. The value for the quadrupole deformation parameter necessary to fit the inelastic data was 0.44. This is to be compared with the value of 0.64 adopted by Stelson and Grodzins 26) from reduced electromagnetic transition probabilities. Stelson and Grodzins used a sharp-edge charge distribution, and as is discussed by Owen and Satchler “), one can deduce a larger value for the quadrupole deformation parameter in this case than when a diffuse-edge nuclear shape is used. However, we place a k20 % uncertainty on our measurement of the quadrupole deformation parameter because of combined uncertainties in the experimental cross sections and nuclear, optical-model parameters. The B(E2) measurement used by Stelson and Grodzins 26) has a probable error of 527 %. The two determinations, therefore, agree to within experimental accuracy without even considering the differences due to the use of a sharp-edge 3.2. PICK-UP
charge distribution REACTIONS
versus a diffuse-edge
LEADING
nuclear
shape.
TO zlNe
The pick-up of a neutron from 22Ne leading to the first three levels of 21Ne was studied with two reactions; the (p, d) reaction (results shown in fig. 2) and the (d, t) reaction (results shown in fig. 3). For each type of reaction the only strong transition is the one leading to the first excited state of 21Ne. Howard et al. ‘) have summarized the spectroscopic data on 21Ne. The first three levels have J” = 2+, 3’ and sf, respectively. The distorted wave theory predictions for the pick-up of a j = 3 neutron using the optical-model parameters in table I are shown in figs. 2 and 3 for transitions going to the 0.349 MeV state of “Ne. The spectroscopic factor deduced by fitting the (p, d) reaction theoretical curve to ex-
596
H.
F.
LUTZ
et
al.
perimental data is 1.86. No attempt was made to deduce an absolute spectroscopic factor from the (d, t) reaction because of uncertainties in the optical-model parameters for the outgoing tritons. With the Nilsson model used, the ground state of “Ne would be described in most simple terms as two neutrons in orbit 7, each with $2” = +jf, outside an inert
Fig. 2. Results for the (p. d) neutron pick-up reaction leading to the first three known states of 21Ne. The proton bombarding energy was 18.2 MeV. Note that the scale for the differential cross section is ten times greater for the reaction to the 0.349 MeV state than for the other two states. The spins and parities of these three states are known to be #+, f+, =$+for the ground, first excited and second excited states, respectively. The Nilsson model predicts a strong transition to the 4 only as is seen in this experiment. The curve is the DWBA prediction for pick-up of a Id+ neutron.
60
0
8’1’1’1’20
40
60
00
‘0 CENTER-OF-MASS
I
I
20
40
I
I
60
I
I
80
ANGLE
Fig. 3. Results for the (d, t) neutron pick-up reaction leading to the first three known states of 21Ne. Note that the scale for the differential cross section is ten times greater for the reaction to the 0.349 MeV state than for the other two states. The curve is the DWBA prediction for the pick-up of a Id+ neutron.
core. The first three levels of “Ne belong to a rotational band built upon an intrinsic state consisting of a single particle in orbit 7. The spectroscopic factor for pick-up of a nucleon from a doubly even, deformed target nucleus is S = 2(Nlj)
Lk~a)~,
(7)
=Ne+d
where the (NZj 1 iha)
AND
=Ne+p
597
REACTIONS
are given by the Nilsson
calculation
3). They are functions
of Nilsson’s deformation parameter q, well-flattening parameter /J and spin-orbit strength parameter K. Harmonic oscillator wave functions are employed in the Nilsson calculations used in this paper. Note that N refers to the number of oscillator quanta in a shell, and in the region of interest the spherical-well orbits Id,, 2s, and Id, all have two oscillator quanta. Nilsson restricts his calculation to a single oscillator shell. The particular orbit of interest (a = 7) has only Id, and Id, spherical-well components with (22% [3+7)’ taking on the value of 1.0 at q = 0 and 0.94 at q = 6. The values that (22% I 3+ 7)’ assumes are 0.0 at u = 0 and 0.06 at q = 6. These considerations predict a strong j = *,I = 2 transition from the ground state of 22Ne to the first excited state of “Ne, and little or no pick-up going to the ground and second excited states of 21Ne. The results of both the (p, d) and (d, t) pick-up reactions are consistent with the above description of the wave functions for the lowlying levels of 21Ne and the ground state of “Ne. 3.3. STRIPPING
REACTIONS
LEADING
TO 23Ne
Ten levels of 23Ne were analysed in the (d, p) stripping reactions. These are all .’Levels up to and including the 3.99 MeV level reported by Freeman 5). The angular
IO2
30 GROUND
‘0
30
60
MeV
STATE
STATE
90
120
150
180
30
60
CENTER-OF-MASS
90
120
150
I80
120
150
180
ANGLE
Fig. 4. The (d, p) stripping results for the reactions leading to the first six known states of 23Ne. The curves are the DWBA predictions calculated with the optical-model parameters listed in table 1 of the text. The spectroscopic factors deduced by normalizing DWBA curves to experimental data are summarized in table 2.
distributions and theoretical curves are displayed in figs. 4 and 5. Two transitions, one going to the 1.70 MeV state and the other to the 2.52 MeV state, exhibit no stripping patterns. The reaction leading to the 1.83 MeV state has been interpreted in the present experiment as a weak j = 3 transfer, although an experiment by Pullen
H.
598
F.
LUTZ
et d.
et al. 6, at a lower bombarding energy (Ed = 7.0 MeV) interpreted this transition as having no stripping pattern. For the low-lying levels reached by I = 2 transitions, it was possible to distinguishj = 3 fromj = 3 transitions by the method given by Lee and Schiffer 30) for I= 2 transitions in the ld-2s shell. The angular distributions displayed in fig. 5 proceeding to the levels of 23Ne at 3.22, 3.43, 3.84 and 3.99 MeV all indicate peaking at O”, making the assignment of l-values less definitive than for the low-lying excited states. The 1 = 2 transitions can
30
**Ne(d p)23Ne 3.43 t&V STATE JZ =2 j=3/2
/I I. LW
90
3.84 L=I
120
150
180
MeV STATE j=3/2
120 150 180 CENTER-OF-MASS
3.99 8=2
MeV
STATE j=3/2
ANGLE
Fig. 5. The (d, p) stripping results for the reactions leading to the seventh through tenth known states of 2SNe. As discussed in the text, there exists other experimental evidence that the 3.22 MeV state has .P = &+. The difference in shape between I = 0 and 1 = 1 DWBA curves is not sufficient at the particular bombarding energy and Q-value to make the assignment unambiguous.
be distinguished from the 1 = 0 and 1 = 1 transitions, but the latter two cause confusion. The assignments made for the 3.22 MeV and 3.84 MeV states were made on rather general expectations of the level sequence expected for negative-parity states and the expected strengths of the transitions populating them. This point will be discussed in detail after the section on the RPC calculation. Pullen et al. 6, have repeated their experiment at Ed = 4.0 MeV and have assigned 1 = 1 to the transition going to the 3.22 MeV level. The work of Howard et al. 8, seems to corroborate this
=Ne+d
assignment.
To indicate
AND
the existence
=Ne+p
REACTIONS
of this contradictory
599
evidence,
we have placed
parentheses around our spin and parity assignment for this level. It should be emphasized that all the spin and parity assignments discussed in this section were made independently of the work of Pullen et al. 6, and Howard et al. *). Also, as will be discussed later, the assignment of I = 1, j = 3 to the 3.22 MeV level creates even more difficulty in interpreting 23Ne within the framework of the Nilsson model. The distorted wave Born approximation code of Macefield 31) was used to generate the theoretical curves for the stripping angular distributions and to extract the spectroscopic factors by comparison with experimental results. These calculations were made on the IBM 7030 (STRETCH) computer at Livermore. The optical model parameters used are given in table 1. The deuteron optical model parameters were found by the search program described in subsect. 3.1. The proton optical model parameters were calculated from the formulae of Perey 28) that account for a large quantity of proton scattering data in the energy region of interest. It was assumed for the calculation of the proton optical model parameters that 23Ne had a quadrupole deformation parameter of 0.5. The cross section can be written
(8) where the factor 1.48 comes from the use of the Hulthtn wave function for the deuteron. Macefield’s code was used to calculate (21,+ 1)/(2ri + 1)0(6),,,. , and comparison with experimental data gave the spectroscopic factor S. Two sets of stripping calculations were performed which differed from one another only in the manner of treating the bound-state neutron wave function. In the first set, the radius parameter was varied while the depth of the well was kept fixed at 45 MeV. In the second set of calculations, the eigenfunction was generated in a real potential well with the Woods-Saxon shape whose size was determined by R = 1.25 A’ and whose depth was varied until the correct binding energy was obtained. Pinkston and Satchler 32) h ave stressed the importance of using a single-particle wave function with the correct asymptotic form and have stated that both the abovementioned procedures are equally correct. In this paper the two procedures are used to ascertain the degree of uncertainty in the determination of the spectroscopic factors due to the uncertainty in describing the correct wave functions for the captured neutron 3 “). The spectroscopic factors deduced by DWBA analysis of the 22Ne(d, p)23Ne stripping reactions are presented in table 2. For the ground state of 23Ne, VN = -45 MeV when rn = 1.25 fm. This fortuitous situation resulted in no difference between the two procedures for handling the bound-state neutron wave function in the case of the transition to the ground state. For the other transitions, the larger value of the spectroscopic factor was usually deduced for the case when rN = 1.25 fm and the potential depth VN varied. The single exception is the level at 2.31 MeV.
600
H. F. LUTZ et d.
Note the very large error due to uncertainty in treating the bound-state wave function for the transition going to the negative-parity state at 3.84 MeV excitation energy. This transition selects out the 2 pq component of the wave function for the 3.84 MeV level. Why this level should occur so low in the spectrum is discussed in the following section. The uncertainty in treating this level which is displaced in energy from its normal spherical shell-model location is the same type of uncertainty encountered by Sherr et al. 34) in th eir study of (p, d) reactions to isobaric analogue levels. The problem has been discussed in detail by Pinkston and Satchler 32). TABLE
Spectroscopic
QWeV)
.WMeV)
2
factors deduced for eeNe(d, p)2SNe reactions Nlj
s
A(%) -
0.0
2.96
0.23 0.23
1.02
1.94
1.83
1.13
2.31
0.65
0.37 0.40 0.020 0.026 0.077 0.064
3.22
-0.26
3.43
-0.47
3.84
-0.88
3.99
-1.03
Remarks for this level only, V, = -45 MeV when rN = 1.25 fm
5 15
0.64 0.81 0.35 0.40 0.04 0.24
9
for this level only S, obtained with V, = -45 MeV
14
other experiments have assigned Jn = t- to this level
9 71
negative-parity state belonging to band based on Nilsson orbit 14
5
0.30 0.33
As explained in the text, two sets of calculations were performed which differed in the manner of treating the bound-state neutron wave function. In this table, the first entry in the column for S is the case when r~ varied and V, was held constant; the second entry is for r, constant and V, varied. The following definitions are used: A
where S< (S,) calculations.
=
@-‘<)
-
x
‘00
S
is the larger (smaller) value of the spectroscopic
factor resulting
from these
4. RPC calcula@on for 23Ne energy levels and spectroscopic factors The expression derived by Satchler 9, for the spectroscopic factor for stripping of a nucleon leading to a nucleus described as an odd-nucleon strongly coupled to an axially-symmetric deformed core is
22Ne+d
where g is initial and assumed to describing
AND
=Ne+p
601
REACTIONS
42 for a doubly even target nucleus and (Q2 I Q1> the overlap of the final vibrational parts of the total wave function. This overlap is usually be unity for transitions to low-lying levels. The (NZj 1 QOCY)are coefficients the deformed-well eigenfunctions in terms of the spherical-well, shell-
model wave functions IQC0a) = c (NZj~GBa>~NZj),
(10)
Nlj
where the summation is shell, N is 2. The symbol the symmetry axis of the The coefficients obey the
usually restricted to a single oscillator shell. For the ld-2s Q is the projection of the total spin of the odd nucleon onto potential well, w the parity and a the Nilsson orbit number. sum rules &(NZjlQwa>’
The Hamiltonian that rotation-particle coupling
= 1,
(11)
G~~(NZjlQwa)z = 2j+l.
(121
describes is
H = Hintr+
the rotational
$(Z’-21;jb)+
motion
$(l;jl,+ri,
of the nucleus
j;),
and the
(13)
Hi,,, is the part of the Hamiltonian describing the intrinsic states and has been solved by Nilsson 3, with the input parameters (q, p, JC). The primes denote the fact that the spherical components of I and j are referred to body-centred coordinates. It is convenient to allow A, = h2/29= where the subscript a refers to a particular intrinsic state. Neglecting the last term of eq. (13) gives the basic set of wave functions IIMKa)
=
[
1
f{D~,lKa)+(-l)r-‘D~-,l-Ka)~,
$$
where the DLK are rotational wave functions, and a is necessary to label different intrinsic states with the same value of K. The [Ka) are Nilsson wave functions defined as in eq. (10). The last term in eq. (13) is responsible for coupling states with the same value of I with AK = f 1 or AK = 0 if, and only if, K = K’ = +. The diagonal elements of the Hamiltonian are E(NlKa)
= E~+A,{[I(I+I)+a(-1)‘+~(I+~)&3]-[K(K+1)-a6K~]},
(15)
where E:: = E,, + A,[K - KZ - u&~] ; and the decoupling
parameter
(16)
is
a = - c (NZjlS2coa)2(j+$)(-l)j+*.
(17)
II.F.
602
The off-diagonal
elements
LUTZ
of the Hamiltonian
I acrra = (IMK’a’J2(1;
.?t al.
are
jl_, +I’_, j;lzMKa)
= -JU(Z+K,)(Z-K>+l) X[1-6K’K+(-1)r-~~Kf8K)]a.‘.,
(18)
where K, is the greater of K’, K, and A is [+(A, + A,)]; a,,a = 7 (N2jl~oa’>
matrix
is diagonalized
(19)
to give the wave functions
IZM) = 1 CJZMKa) Ka
and the eigenvalues
for each mixed state with spin I. With band mixing,
expression
(9) becomes
(21) The effects of pairing
are included
JSlj =
by writing
[A]’ f
E G, U,(NMoa),
(22)
where U,” is the probability that a particular state is filled 14). The off-diagonal elements are modified by the effects of pairing by multiplying each off-diagonal element by (U,. U,+ V,. V,), where 17, and V, are related by
u,z+v,z = 1.
(23)
described A computer code developed by Erskine 35) performed the calculations above. The input data for the calculation is given in table 3. The RPC calculation (results summarized in table 4) included levels belonging to rotational bands built on five intrinsic states that could be expected, from inspection of the Nilsson level diagram, to make contributions to the positive-parity levels of 23Ne considered in this paper. Efforts were made to minimize the number of parameters varied. The A, associated with the amounts of inertia are similar in value to others used in calculations in this region of the nuclear mass table. The parameters U, were chosen to produce the effects of pairing in at least a qualitative fashion. The values for U, are felt to be reasonable, but they are not the result of any further calculation. The outstanding differences that resulted between experiment and the calculation were not dependent on the choice of U,. This was ascertained by performing the calculation without considering hole states. Various calculations were made for q increasing in steps of 0.5. The value 3.0 was the largest value that could be used with harmonic oscillator wave functions since, for larger values of q, the Nilsson orbit 9 falls at a
“Ne+
d AND
“Ne +
603
p REACTIONS
TABLE 3 Input data for RPC calculation with pairing corrections Orbit number a
Asymptotic quantum numbers Nn,A Li
Energy from Nilsson level (MeV)
202 +
5
Base energy (MeV)
(I&)
uz,
0.
0.
0.3
0.5
9
211:
0.39
0.66
0.15
0.8
7
211%
2.26
1.46
0.15
0.2
11
200 &
2.94
2.82
0.15
1.0
8
202 #
3.98
3.48
0.15
1.0
Nilsson wave functions were generated with 17= 3, ,U = 0.2 and K = 0.06. The decoupling parameter for orbit 9 is -0.2445 and for orbit 11 is -0.2817.
TABLE
4
Results of RPC calculation for positive-parity levels showing spins, energy levels with Coriolis band mixing considered, the strengths of transitions with and without band mixing and the wave functions as defined by eq. (19) of the text CKca
@J-t 1 )S
QJ+
CK,
CK,
cK,
CK,
1)s
K=@
K=:
K=f
K=:
a=5
a=9
a=7
a=
K=$
11
a=8
(lZV)
mixed
unmixed
1.07 3.27
0.67 0.89
0.52 1.04
-0.995 0.097
1.25 1.93 3.56 4.07
1.42 0.24 1.43 0.58
0.83 0.01 0.90 0.94
-0.942 0.227 0.246 -0.034
-0.250 -0.966 -0.069 -0.014
-0.180 0.115 -0.859 -0.465
-0.135 0.054 -0.443 0.885
0 2.05 3.13 4.16 5.29
1.44 0.28 0 0 0.03
1.0 0.25 0.39 0.06 0.06
0.924 0.162 0.342 0.050 -0.021
-0.060 0.922 -0.266 0.039 0.273
-0.316 0.232 0.893 0.089 -0.007
0.016 -0.223 -0.002 0.724 -0.653
-0.032 0.143 -0.124 0.681 -0.706
1.33 2.62 4.12 5.55 6.17
0 0 0 0 0
0 0 0 0 0
-0.796 0.200 0.526 -0.218 0.049
0.168 0.879 0.103 0.431 -0.054
0.577 -0.023 0.736 -0.353 0.002
0.021 0.342 -0.337 -0.693 -0.536
0.069 0.264 -0.241 -0.403 -0.840
2.84 4.37 7.37 8.40 6.22
0 0 0 0 0
0 0 0 0 0
0.644 0.287 0.686 -0.118 -0.136
-0.272 0.883 -0.035 0.375 0.072
-0.704 -0.139 -0.673 -0.091 -0.155
0.092 -0.286 0.273 0.640 0.652
-0.085 0.194 0.030 -0.654 0.726
0.097 0.995
604
H.
F.
LUTZ
et
cd.
lower energy than does orbit 5. This would give a ground state J" = +' instead of 5 + as is known from experiment. The value (0.2) taken for p is the one suggested by &shop 36). The value of K determines the d,-d, splitting. It was found that IC = 0.06
-
aii
;-r -
1
‘\-’
-\
I I-
-’ ---
i -
1 0t -\
'K-5/2 a-5
-\= ;;2 a.9
_I’ -.-
-
--K*l12 cr.11
K=3/2 a.9
K=3/2 a-7
Fig. 6. The effect of the Coriolis band mixing on the position of the positive parity levels considered in the RPC described in the text and summarized in tables 3 and 4. For each band, the level sequence without Coriolis band mixing is shown on the left-hand side, and then the position of the levels after the Coriolis interaction is considered is shown on the right-hand side. The ground state is depressed by several hundred keV. This is true only if the hole in orbit 7 is included in the calculation. THEORY
0
EXPERIMENT
I (2J +I)S
Fig. 7. Comparison
2
0
I (2J+I)S
2
of the results of the DWBA analysis and RPC calculation for the positive-parity levels of 2SNe.
would place orbit 8 near 4 MeV excitation energy, as indicated by the stripping results. The values of the decoupling parameters for orbits 9 and 11 were calculated from the Nilsson wave functions generated for the intrinsic states with the chosen set of parameters (q, p, K). They were not treated as parameters to be varied. The base
=Ne+d
AND
-=Ne+p
605
REACTIONS
energies for the rotational bands were varied slightly from the Nilsson level values. The effects of the Coriolis band mixing on the position of the positive parity levels is shown in fig. 6. For each band, the level sequence without Coriolis band mixing is shown on the left-hand side, and then the position of the levels after the Coriolis interaction is considered is shown on the right-hand side. Note that the ground state is depressed by several hundred keV. This is due largely to the Coriolis interaction between ground state (1 = 4, K = $) and the second member of the band built on orbit 7 (I = 4, K = 3). Orbit 7 must be included in the calculation to depress the ground state and give it a K = $ component. The results of the RPC calculation for the positive parity levels of 23Ne are summarized in table 4. The results of the DWBA analysis of the experimental data and the RPC calculation are compared in fig. 7. The RPC calculation predicts that we should see eight transitions with (2J+ 1)s greater than 0.1 in the energy region scanned by this experiment proceeding to positive-parity states. The data reveal at most seven. (If the 3.22 MeV level is negative parity, then only six are seen.) The TABLE
Results
of RPC calculation
for negative-parity
5
transition
belonging
(2J+ 11.9 mixed
4.87 3.62 5.84 3.75
0.07 0.52 0.18 3.39
to band
built on Nilsson
orbit 14
(2J+ 1)s unmixed
0.06 0.48 0.04 1.41
The Nilsson parameters were 7 = 4.5, ,u = 0.35 and K = 0.06. The parameter A, was taken to be 0.150 MeV. The base energies for the bands were taken from the Nilsson calculation. The decoupling parameter was -3.205.
missing one is a strongj = 3, 1 = 2 transition to a state above the first J” = 4 level. Some RPC calculations were performed by using only four intrinsic states and omitting the hole-state in Nilsson orbit 7. The weak j = 3 transition to a level around 2 MeV excitation energy was deleted in this fashion, but the predictions of this truncated calculation for the other seven transitions agreed qualitatively with the predictions of the full calculation. Especially evident was the prediction of a strong transition, (2J+ 1)s = 2, going to a 3’ level near the first rlf level. In fact, it was found that this transition is predicted whether one considers orbit 7 or not and whether or not Coriolis band mixing is included. One possible explanation is suggested by examination of fig. 6. The 4 and f members of the K = 3 band built or orbit 9 are pushed together by the Coriolis band mixing. It is possible that they may actually be so close in energy that they cannot be resolved experimentally. The DWBA fit to the angular distribution for protons going to the J” = &+ state at 1.02 MeV could be improved by adding a j = 3, 1 = 2 angular
H.
606
F.
LUTZ
et
al.
distribution of about the magnitude suggested by the RPC calculations. (A strength (2J+ 1)s of 1.4 implies a peak cross section of about 7 mb/sr.) The comparison of theory and experiment suggests that the two levels we see that do not exhibit the usual stripping pattern are the 3’ members of the ground-state band and the band built on orbit 9. An RPC calculation was also performed for the negative parity states of the N = 3, lf-2p shell. All ten intrinsic Nilsson states for the N = 3 shell were included in the computation made for r] = 4.5, p = 0.35 and rc = 0.06. The parameters A, were all taken to be 0.150 MeV and the positions of the band heads taken from the Nilsson calculation. With the above mentioned values for the Nilsson parameters (q, ,u, rc), the decoupling of orbit 14 was calculated to be -3.205. This is large enough to produce inversion of the ordering of the levels for the band. The results are summarized in table 5. One expects to see a level sequence 3-, I-, &- and 8- with strong transitions going to the $- and 3- levels. If the 3.22 MeV state has J” = -&-then two 1 = 1 transitions are seen in the region studied. The level sequence for a decoupling parameter in the range - 4 to - 3 (for orbit 14 this corresponds to q varying from 0 to 6) is +,$,t, 3 implying that an 1 = 3 transition should occur between the two 1 = 1 transitions. Furthermore, if we fit the 3.22 MeV angular distribution with an 1 = 1,j = 3 DWBA curve, the value deduced for (2J+ 1)s is about 1.7. This is about three times larger than any I = 1 transition listed in table 5. 5. Summary The neutron pick-up and stripping reactions studied in this paper have investigated the low-lying levels of ‘rNe and 23Ne. The transitions to “Ne populate the ground state rotational band and agree well with the predictions of the Nilsson model. The RPC calculations for the structure of 23Ne, as employed in the present paper, are not expected to give detailed agreement with experimental data; but the discrepancies revealed by the stripping reactions leading to levels of 23Ne disclose certain major discrepancies between the experimental data and the theoretical results. These discrepancies consist of strong transitions predicted theoretically but not seen experimentally, and they persist whether or not one considers the hole-state in orbit 7 and whether or not Coriolis band mixing is included. The authors would like to express their sincere thanks to Dr. J. R. Erskine for the use of his RPC computer code and for several useful communications, to Dr. B. E. F. Macefield for his DWBA code, to Dr. R. Barrett for his adiabatic -approximation scattering code and Dr. D. J. Pullen and Dr. A. J. Howard for receipt of data prior to their publication. Note added in prooJ
bound-state
Dr. Satchler has informed the authors that, in treating the wave function, Pinkston and Satchler 32) suggested that varying the
“Ne+
d AND
B2Ne+p
REACTIONS
607
radius of the potential might be more physically correct than varying the depth and that both procedures give the correct asymptotic tail. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 3 1) 32) 33) 34) 35) 36)
G. Rakavy, Nuclear Physics 4 (1957) 375 R. K. Sheline and R. A. Harlan, Nuclear Physics 29 (1962) 177 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) A. E. Litherland et al., Can. J. Phys. 36 (1958) 378 J. M. Freeman, Phys. Rev. 120 (1960) 1436 D. J. Pullen, A. Sperduto and E. Kashy, Bull. Am. Phys. Sot. 10 (1965) 38 A. J. Howard, D. A. Bromley and E. K. Warburton, Phys. Rev. 137 (1965) B32 A. J. Howard et al., Bull. Am. Phys. Sot. 11 (1966) 406 G. R. Satchler, Ann. of Phys. 3 (1958) 275 J. R. Erskine, Phys. Rev. 138 (1965) B66 A. K. Kerman, in Nuclear reactions, Vol. I, ed. by P. M. Endt and M. Demeur (North-Holland Publ. Co., Amsterdam, 1959) chapt. X H. Lancman, A. Jasinski, J. Kownacki and J. Ludziejewski, Nuclear Physics 69 (1965) 384 D. Kurath, private communication S. Yoshida, Phys. Rev. 123 (1961) 2122 E. Ye. Berlovich, Phys. Lett. 13 (1964) 161 G. Alaga, K. Alder, A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 9 (1955) P. E. Hodgson, The optical model of elastic scattering (Oxford at the Clarendon Press, 1963) D. M. Brink, Proc. Phys. Sot. A68 (1955) 994 S. Yoshida, Progr. Theor. Phys. 19 (1958) 169 W. Tobocman, Theory of direct nuclear reactions, (Oxford University Press, 1961) S. K. Penny and G. R. Satchler, Nuclear Physics 53 (1964) 145 H. P. Hernandez, J. M. Peterson, B. H. Smith and C. J. Taylor, Nucl. Instr. 9 (1960) 287 F. S. Goulding, D. A. Landis, J. Cerny and R. H. Pehl, Nucl. Instr. 31 (1964) 1 R. N. Maddison, Proc. Phys. Sot. 79 (1962) 264 R. C. Barrett, Nuclear Physics 51 (1964) 27 P. H. Stelson and L. Grodzins, Nucl. Data Al (1965) 21 L. W. Owen and G. R. Satchler, Nuclear Physics 51 (1964) 155 F. G. Perey, Phys. Rev. 131 (1963) 745 J. R. Rook, Nuclear Physics 61 (1965) 219 L. L. Lee, Jr. and J. P. Schiffer, Phys. Rev. 136 (1964) B405 B. E. F. Macefied, private communication W. T. Pinkston and G. R. Satchler, Nuclear Physics 72 (1965) 641 J. R. Erskine et al., Phys. Rev. Lett. 14 (1965) 915 R. Sherr, E. Rost and M. E. Rickey, Phys. Lett. 12 (1964) 420 J. R. Erskine, private communication G. R. Bishop, Nuclear Physics 14 (1959) 376