- Email: [email protected]
The 22Ne(p, t)20Ne reation at 26.9, 35.1 and 42.4 MeV
I
2-F
I
Nuclear Physics A167 (1971) 157-176; Not to be reproduced
by photoprint
@ North-Holland
Publishing
or microfilm without written permission
THE 22Ne(p, t)“Ne
Co., Amsterdam from the publisher
REACTION
AT 26.9, 35.1 AND 42.4 MeV W. R. FALK, P. KULISIC + and A. MCDONALD University
of Manitoba,
Winnipeg,
Received 30 December
Canada ++
1970
Abstract: Angular distributions of tritons from the 2ZNe(p, t)20Ne reaction have been measured at proton bombarding energies of 26.9, 35.1 and 42.4 MeV. All the known energy states up to an excitation energy of 6.72 MeV in “‘Ne were excited, including the unnatural parity 2- level at 4.97 MeV. DWBA calculations have been made using the finite-range two-nucleon transfer code of Nelson and Macefield. Various wave functions describing Z”Ne and 22Ne, taken from the work of Arima et al., Wildenthal et al., Kuo and Zuker, were used in calculating the spectroscopic amplitudes for transitions to the various final states in 20Ne. Comparison of the nuclear structure information extracted at the different bombarding energies is emphasized. The sensitivity of the DWBA results to the binding energy of the singleparticle states is discussed. E
NUCLEAR
REACTIONS
“Ne(p,
t), E = 26.9, 35.1, 42.4 MeV; measured
o(e),
1. Introduction
Most studies of the two-nucleon (p, t) transfer reaction have been performed at only one value of the proton bombarding energy. The extraction of nuclear structure information from such studies involves numerous uncertainties, one of which is the energy dependence of the kinematical aspects of the reaction. In an attempt to check the reliability with which the nuclear structure information can be extracted in two-nucleon transfer reactions, the 22Ne(p, t)20Ne reaction was investigated ‘) at proton bombarding energies of 26.9, 35.1 and 42.4 MeV. The level structure for low-lying states in “Ne is well known 2- “) and h ence only transitions to levels of known spin and parity were involved. Several descriptions of nuclei in the sd shell within the restricted shell-model space of the ld,--2s, orbits have recently been presented 6*‘). Such descriptions are found to predict satisfactorily for example, the data on spectra, transition rates and the static multipole moments. In both the work of Arima et al. “) and Wildenthal et al. ‘) the two-body matrix elements were determined by adjusting them to minimize the rms deviation between the calculated and experimentally observed energy levels. Only the energy levels of nuclei with two to four nucleons outside the 7 Present address: Institute R. BoSkovid, Zagreb, Yugoslavia. tt Supported in part by the Atomic Energy Control Board of Canada. 157
158
W. R. FALK el al.
’ 6O core were considered in deducing the matrix elements in the former investigation, whereas in the latter, energy levels of nuclei with A > 20 and N, 2 < 14 were taken into account. More elaborate shell-model descriptions of nuclei in this mass region have been presented by Kuo “) and Zuker “). The latter considers the Ip, as well as the Id+ and 2s+ orbits, although the Id, orbit is omitted. Within this framework a description of the negative parity states is thus also provided. Kuo’s “) shell-model calculation encompasses the entire sd shell. However, in calculating the spectroscopic information for the DWBA calculations it was necessary to truncate his results omitting the Id, orbit, since the shell-model computer program was restricted to handling only two shell-model orbitals. The foregoing shell-model descriptions of the nuclei “Ne and 22Ne were utilized in determining spectroscopic amplitudes for the DWBA calculations for the 22Ne(p, t) 20Ne reaction.
2. Experimental The experimental has been described
procedure and results
layout of the University in an earlier publication
of Manitoba
sector focused
cyclotron
’ “). The overall experimental arrangement and data-taking procedure was the same as that used for the 20Ne(p, t)18Ne reaction lo) except that 3He and cr-particles were not recorded in the present experiment. Neon gas (obtained from Monsanto Research Corp.), enriched to 99.6 % 22Ne was contained in a 6 cm diameter gas cell at a pressure of about one atmosphere. The pressure in the gas cell was monitored by observing the elastically scattered protons in two NaI scintillation counters placed at 373” on either side of the proton beam. Entrance and exit windows from the gas cell consisted of 25.4 pm Kapton-H foil (manufactured by Du-Pont) which resulted in negligible energy straggling of the outgoing tritons. Triton spectra recorded at 26.9 and 42.4 MeV proton bombarding energy are shown in fig. 1. All the low-lying states -O.O(O+), 1.63(2+), 4.25(4+), 4.97(2-), 5.63(3-)+ 5.80(1-) and 6.72(0+) MeV states-are clearly observed in the spectra. In addition, numerous more highly excited states, particularly at 42.4 MeV bombarding energy, are visible in the spectra. The overall energy resolution obtained varied between 140 and 170 keV, the largest contribution of which was due to the energy spread of the incident proton beam. Beam currents between 10 and 80 nA, depending on the angle of measurement, were used in recording the data. Angular distributions for the observed states are shown in fig. 2 at the three measured energies. Both the L = 0 (0.0and 6.72 MeV states) and L = 2 (1.63 MeV state) angular distributions retain their characteristic shapes despite the variation in outgoing triton energies from approximately 11 to 33 MeV (ground state Q-value equals -8.664 MeV). The other angular distributions show greater variations in
159
22Ne(p, t)‘ONe
shape particularly at the lowest energy of 26.9 MeV. Efforts were made to ensure that the errors in the relative cross sections for the different energies were minimized. Toward this end the same detector geometry was used throughout and comparative 15022
EP
Ne(p,tj2’Ne =269
MeV
‘1 AB = 35”
CHAN
N EL
~0.
22Ne (p,t)20Ne Ep = 42 4 MeV
CHANNEL
Fig. 1. Triton
spectra
recorded
0
600
500
400
NO
at 26.9 and 42.4 MeV proton
bombarding
energy.
measurements at 42.4 MeV at the beginning and end of this series of experiments were made. The results from these latter measurements were the same within the statistical error limits. Error bars shown in fig. 2 represent the statistical counting
160
W. R. FALK ei al.
IQ00
l
26.9
NIV
.
33.1
MCV
D
42.4
NIV
800
200
40
0
0
20
40
60
80
0’
1 I
0
,
20
1
I
40 8 c-m.
I
,
!
60
Fig. 2. Angular distributions of tritons for statajs in *#Ne from the 2ZNe(p, t)2W8 raaction. solid Iines arc smooth lines drawa thrcrugh the data points.
I
80
The
161
22Ne(p, t)‘ONe 240-
22
NCIP.l P [email protected]
1
“Ne
(P.t?“N~(672MeV,0+I
and , 80 MeV.I- 1 026.9
200-
Ye”
n 35.1
MIV
35.1
M.”
0 42.4
Yev
0 42.4
Ye”
40-
160-
z
McV
.26.9 .
< 0 5
120-
q “j
30-
2: I
414 60’
20-
40-
10
0;
I
0
10
I
20
30
40
50
60
70
Of
00
I 0
I
IO
0 c.m.
r 20
I
30
I
I
I
40
50
60
I
70
I
80
e cm
Fig. 2 (continued).
errors only; surements.
an overall
normalization
uncertainty
of + 10 % applies
to the mea-
3. DWBA calculations 3.1. DWBA FORMALISM
A summary of the DWBA formalism for two-nucleon transfer reactions was presented in an earlier paper lo). For completeness, this summary is repeated here. Following the treatment given by Towner and Hardy 11) we write for the DWBA expression
for the differential
cross section
A(a, b)B:
do - = .&!!C 5 dQ (27~rh~)~ k,
x
c
[“111j~‘[nzz*i23
S:&i,
1, jl][n2
I,
j,];
JT)
where [nr Z,j,] and [n,Z,j,] are the quantum numbers defining the single particle states of the transferred nucleons, and L, S, J and Tare the quantum numbers of the transferred pair taken as a cluster. All the nuclear structure information is contained in the factor Sin, called the spectroscopic amplitude, D(S, T) is a quantity depending on the spin and isospin dependence of the nucleon-nucleon interaction and b&- is essentially a spectroscopic factor for the light particles. The amplitudes Bi
162
W. R. FALK
et al.
describe the kinematical aspects of the reaction. It is clear from the sum over [nl Z,j,] and [Q I2j,] appearing inside the brackets that different configurations for the two nucleons can contribute and these are added coherently. The nuclear structure information can thus not be factored from the nuclear reaction dependence in twonucleon transfer reactions as in the case of single-nucleon transfer reactions. In the expression (3.1) spin-orbit forces have been neglected. The spectroscopic amplitude Sk, is a measure of the overlap of the wave functions of the target nucleus and the final nucleus plus two nucleons defined by quantum numbers [n, I, j,] and [a2Zzj,]. Formally it is defined as
x IcI$$+“&~, ~1, rz) d& du, dr, .
(3.2)
Here tB refers to the coordinates of the A -2 nucleons in nucleus B, and the square brackets within the integral sign denote vector coupling. Given a certain model, say the shell model, for which the wave functions can be determined for the nuclei A and B, the spectroscopic amplitudes can be calculated. Such calculations were carried out for the “Ne(p, t)“Ne reaction for different wave functions describing 22Ne and “Ne as deduced by various investigators. These were described in some detail in the introduction. The actual calculation of the spectroscopic amplitudes was performed by Towner 12) using the shell-model program of Towner and Davies 13). Results of these calculations are shown in
Two-nucleon
State
Exp. ext. energy
spectroscopic
02 +
0.00
6.12
Arima (lP# (ld;)’ (2s+Y
-0.866 -0.168
(lP+Y (Id+)’ (2s$
-0.050 -0.075
11-
5.80
(lP+2@
21 +
1.63
(ld# (ldz2s+)
31-
5.63
(lp+ldz)
41 +
4.25
reaction
Configuration
(MeV)
01+
TABLE 1 amplitudes for the ZZNe(p, t)‘ONe
(Id*)’
“)
Wildenthal
-0.943 -0.282
“)
Kuo b,
-0.859 -0.304
Zuker
‘)
0.246 1.021 0.206 0.020 0.062 0.083
0.001 0.118
0.017 0.214
-0.314 0.094
0.328 -0.224
-0.212 0.297
-0.704 -0.090
- 1.279
- 1.084
-1.130
-0.651
-0.231
1.212
“) Relative signs of the sd components of the wave functions have been changed (see text for explanation). b, The Id+ orbit has been omitted from Kuo’s results prior to calculation of the spectroscopic amplitudes. ‘) Calculations limited to configurations of lowest seniority, i.e. v S 2.
22Ne@, t)*‘Ne
163
table 1 for various final states in 2oNe. As noted ea rli er , excep t for the wave functions of Zuker “) only the Id+-2s, orbits were considered; the Id, orbit was omitted from the model space prior to calculation of the spectroscopic amplitudes using Kuo’s “) matrix elements. A point of uncertainty arises in regards to the spectroscopic amplitudes obtained from the wave functions of Arima et al. “) and Wildenthal et al. ‘). In both cases the two-body matrix elements, determined from a fit to the experimental energy levels of sd shell nuclei, leave the relative signs of the sd components of the wave function undetermined. For this reason calculations were also carried out with the relative signs between the s and d components changed. It was found that only with this sign change could reasonable results be obtained. 3.2. DWBA
COMPUTER
PROGRAM
The DWBA two-nucleon transfer code used in the present calculations was that of Nelson and Macefield 14). While no attempt will be made to describe the various features of this code in detail, several pertinent aspects should be mentioned. A finite range option is incorporated in this code that allows the Gaussian parameters of the interaction range and the triton size to have different values. For all the calculations employing the finite range option an interaction range of 1.6 fm was adopted. The triton size parameter was fixed at 0.240 fm-‘. Several methods of calculating the wave function of the transferred two-nucleon pair have been presented 11,15). In the Glendenning theory 15) the two transferred nucleons are described by harmonic oscillator wave functions for which a separation into functions of the relative motion and the motion of the center of mass of the pair can readily be performed using the Moshinsky transformation 16). The incorrect asymptotic form of the harmonic oscillator wave function for the c.m. motion is replaced by a Hankel function. This calculation assumed a zero interaction range but takes into account the finite size of the triton by using a Gaussian wave function for this particle. An alternative to the above procedure allowing much greater flexibility is to describe the transferred particles by a product of Saxon-Woods wave functions with different radial arguments. The binding energy of each nucleon can then be independently varied. DWBA predictions employing the different methods of calculating the wave function of the transferred pair will be presented and discussed in subsect. 3.3. Further information regarding the details of these calculations can be found in refs. 14*17). The magnitudes of the cross sections predicted by eq. (3.1) are found to be very sensitive to a number of factors entering the DWBA calculation, chief among which are the single-particle binding energies (subsect. 3.3) and the optical-model parameters (subsect. 3.4). Several workers ‘*, ’ “) have investigated this problem but to date there is no adequate description of the reaction that permits reliable predictions of the absolute magnitudes of the cross sections. Consequently, only the relative magnitudes of the DWBA cross sections will be considered in the comparison with the experimental results.
164
W. R. FALK
3.3. SINGLE-PARTICLE
BINDING
et al.
ENERGIES
In the analysis of the “Ne(p, t)“Ne reaction lo) it was noted that variations of the order of 40 % in the predicted cross sections were obtained depending on the bound-state parameters employed. Current theories of two-nucleon transfer reactions do not provide an adequate prescription for calculating the bound state wave functions of the transferred nucleons. Recently Jaffe and Gerace lg) have presented arguments (applicable to ground state transitions of doubly even nuclei with closed neutron and proton shells) which show that the appropriate neutron binding energy to use, for example, in the case of the laO(p, t)160 g.s. reaction, is that of the neutron in “0. This adequately accounts for the pairing energy of the two neutrons. TABLE 2 Summary
of the different
bound-state
Range
parameters Singleparticle well depth
used in the DWBA
calculations
Comments
(MeV) A B C D E F G
“) Oscillator
finite finite finite finite finite finite zero
parameter
6.0 5.5 6.0 5.5 6.0 5.5
55.0 59.0 62.0
59.0
intermediate case of binding (see table 3) binding energy equally divided R = R,,,. = 1.3 fm, A = A,.,. = 0.7gfm Glendenning-type calculation “)
Y = 0.31 frnm2.
In the more general case of nuclei removed from closed shells the situation is much less clear. For this reason a number of calculations were carried out in which the single-particle binding energy was varied. A summary of the particulars of these calculations is presented in table 2. The calculations labelled A, B and C correspond to cases in which the parameters of the potential well were held fixed for calculating lp,, Id, and 2s, single-particle energies. A Saxon-Woods form for the potential was assumed with a radius parameter of 1.20 fm and a diffuseness of 0.70 fm. The spin-orbit potential, V,,,, , varied between 5.5 and 6.0 MeV. The binding energies so generated are shown in fig. 3 for the case of V,,,, = 5.5 MeV. For the well depth of 55 MeV the mean of the d;-s+ neutron separation energies corresponds to the Separation energy of a neutron from ‘lNe . At a potential depth of 62 MeV the mean of the d;-s, neutron separation energy corresponds roughly to one half of the twoneutron separation energy from 22Ne (for the 1.63-4.25 MeV transitions in “Ne). The arrows in the lower portion of fig. 3 correspond to one half of the two-neutron separation energy from ‘*Ne for the 0.0, 1.63,4.25 and 6.72 MeV final states in 20Ne.
22Ne(p,
165
t)*ONe
Calculation E in table 2 represents the opposite extreme of the calculations A, B and C in that the experimental two-neutron separation energy was simply divided equally between the two neutrons, regardless of the orbit from which the neutron was removed. A zero-range calculation, according to the Glendenning formalism, is represented by calculation G. Finally, a choice of single-particle binding energies, intermediate between the extremes represented above, is given by calculation D. Details for the binding energies for the different states for this calculation are given in table 3; essentially, the experimental two-neutron separation energy was divided among the single-particle states taking cognizance of the spin-orbit separation between the two orbits. One further choice of bound state parameters is indicated by
Y 0 40Z v, 2
I
I
I
4
6
8
0.0 1.63 4.25 I
BINDING
10
6.72 I
I
I
I
12
14
ib
18
ENERGY
I
20
22
(MeV)
Fig. 3. Single-particle well depth as a function of the binding energy A spin-orbit potential of 5.5 MeV and radius and diffuseness parameters respectively, were used in the calculations. The positions of the arrows the figure indicate one half of the two-neutron separation energy 1.63, 4.25 and 6.72 MeV final states in ‘ONe.
for a neutron in ‘*Ne. of 1.20 fm and 0.70 fm in the lower portion of in **Ne for the 0.0,
TABLE 3 Representative
single-particle
State J”
02 + 11-
The O+ states
(MeV)
Experimental two-neutron separation energy (MeV)
0.00 6.72 5.80
17.15 23.87 22.95
Energy
01+
binding
are not calculated
for Zuker’s
energies
for calculation Binding
1pt.
D of table 2 energy ld%
2s.t
9.43 13.13
7.72 10.74 4.87
18.08
9, wave functions
(MeV)
in this case.
166
W. R. FALK
et al.
TABLE 4
DWBA
cross-section
predictions
for the different
bound-state
parameters
as defined
in table
2
%xvBA B
A
C
D
E
26.9
01 + 41 +
L=O L=4
(30”) (74”)
40.6 1.36
42.3 0.93
35.1
01 + 41+
L=O L=4
(28”) (60”)
24.3 3.40
26.3 2.58
42.4
0,+ 41 +
L=O L=4
(26”) (50”)
21.8 7.47
15.1 4.73
The overall normalization is arbitrary but the relative cross Spectroscopic amplitudes from the wave functions of Wildenthal are given at the peak in the vicinity of the angle indicated.
11.6 3.39
14.7 3.10
15.9 3.80
F
13.5 4.17
sections are correctly indicated. were used. Cross-section values
calculation F in which the geometric parameters of the Saxon-Woods potential were changed to 1.30 and 0.78 fm (for the radius and diffuseness parameters respectively). An indication of the sensitivity of the magnitudes of the DWBA cross sections for the different calculations A-F described in table 2 is presented in table 4. Spectroscopic amplitudes from the wave functions of Wildenthal (see table 1) were employed in obtaining these results. Variations of the order of a factor of 2 are observed as the binding energy of the nucleons is varied. These variations are much less than those observed in the corresponding “Ne(p, t)“Ne calculations lo) where a large imaginary radius parameter of 1.9 fm was employed (see subsect. 3.4). The ratios of the calculated cross sections, while exhibiting much smaller variations than the factor of two noted above, do, however, still indicate that the choice of the correct bound state parameters is an important aspect of the DWBA calculations. Further discussion of this point will be presented later. 3.4. OPTICAL-MODEL
PARAMETERS
The form of the optical-model
potential
used in the DWBA calculation
is defined by
where j and Vc is the No spin-orbit At the time for the elastic deduced from
(r,,
a,) = [I+exp
{+]I-’
(3.3)
Coulomb potential due to a uniformly charged sphere of radius r&. term was employed. when these calculations were performed no optical-model parameters scattering of protons on “Ne were available. Hence the parameters the p+ “Ne elastic scattering lo) were adopted without modification
“Ne(p, for
these calculations.jThis
t)“Ne
set of optical-model
167
parameters
is shown in the first line
of table 5. For t + “Ne elastic scattering, likewise, there existed no experimental data. Numerous sets of optical-model parameters for the t + “Ne outgoing channel were tried in the DWBA calculations. Finally the set shown in the second line in table 5 was adopted for all the calculations on the basis of a best all around fit to the experimental angular distributions (see fig. 6). Although the energies in both the incoming and outgoing channels vary over a rather wide range (26.9 to 42.4 MeV (lab) for the protons, and approximately 11 to 33 MeV (lab) for the tritons), the same set of TABLE
Optical-model
parameters
used
p+ZZNe t fZoNe Set 1 Set 2 Set 3 Set 4
36.33 150.0 213.0 213.0 150.0 120.0
of the p+2ZNe Imaginary
Real volume
(MZ)
5
for the calculation
volume
and
t+ZoNe
distorted
Imaginary
(2)
(fZ)
(ME&
(f&
(fZ)
Wo (MeV)
1.197 1.54 1.45 1.40 1.50 1.25
0.746 0.66 0.70 0.70 0.60 0.65
11.31 45.0 45.0 45.0 45.0 50.0
1.196 1.54 1.45 1.40 1.50 2.00
0.786 0.66 0.70 0.70 0.60 0.65
0.18 0.0 0.0 0.0 0.0 27.0
waves
surface
(fm”, 1.196
2.00
0.786
0.55
The parameter sets l-3 for t+“‘Ne were used for the calculation of the angular distributions shown in fig. 4. Angular distributions using optical parameters set 4 are shown in fig. 5.
optical-model parameters was used throughout. This approach was adopted since the use of different parameters at the various energies would have introduced many new uncertainties into the interpretation of the results. Justification for such a procedure can be made in view of the general insensitivity of the DWBA calculations to the proton parameters, and the observation that the optical-model parameters for mass-three particles exhibit little energy dependence ““). Somewhat better fits to individual angular distributions were obtained by optimizing parameters for that particular case. Thus the sets of t + “Ne optical-model parameters labelled 1, 2 and 3 in table 5 were found to give improved fits for the cases shown in fig. 4. A markedly different set of optical-model parameters for t + “Ne (set 4 in table 5), with an imaginary radius parameter of 2.00 fm, was also tried. This set was similar in many ways to that used for the t + 18Ne analysis ’ “). However, with such a large imaginary radius parameter, total reaction cross sections are incorrectly determined and absolute values for the (p, t) cross sections are much smaller than reasonable estimates would lead one to expect. With the other sets of t + 20Ne parameters both the total reaction cross sections and the absolute values of the (p, t) cross sections
168
W. R. FALK
et al.
are acceptable. For comparison, DWBA angular distributions calculated with opticalmodel parameters set 4 are shown in fig. 5 for the O:, 0: and 4: states in 20Ne. While somewhat inferior fits are obtained than with the more acceptable t + 2oNe parameters (figs. 4 and 6), the differences are not as great as one might have expected. 3.5. DWBA
RESULTS
The (p, t) reaction on spin-zero target nuclei has the simplifying feature that only one value of the angular momentum transfer L enters into the calculation. Selection m-1
I
22Ne(p,t)‘oNe
42.4 MeV
t
I
I
,
“Ne(p,t)
I
I
2CNe
/
I
42.4 MeV
I 0.0 MeV
-I
1000 -
IOO-
4.25 4:
MeV
Set
2
Set
3
100 -
$ a lo-
1000 $ -0
-
100 -
loo-
-
. ..
l
.
.* .*
.
.
.
IO
IO-
;T:
I IO
Fig. 4. DWBA
I
I I I 30 50 0 cm
I
1 1
70
I IO
I
I 30
I 8
I 50
I
I
cm
fits for the Oi+ and 4,+ states in *ONe using various model parameters
I 70
sets of t+20Ne
optical-
as defined in table 5.
rules, on the basis of a one-step direct pickup mechanism, give the result Jf = L, and restrict the parity to q = (- 1)” = (- l)Jf. The shapes of the angular distributions are largely insensitive to different choices of the bound-state parameters and to the spectroscopic amplitudes.
169
22Ne(p,t)20Ne
DWBA fits to the measured angular distributions (excluding the 4.97 MeV, 2- level) are shown in fig. 6. All calculations for the curves shown were performed with the optical-model parameters shown in the first two lines of table 2, and the spectroscopic amplitudes from the wave functions of Wildenthal et al. ‘) (see table 1). The curves for the 0.0, 1.63, 4.25 and 6.72 MeV states correspond to angular momentum trans6.72 MeV
100
02+ T----J 1
j
~p.269MeV) 100
1
10’
Ep.351
\ .n
MeV]
t
10
10
I I I IO 30
I
8 cm
I 50
I
I 1 70
8 cm
Fig. 5. DWBA fits for the O,+, 02+ and 41f states in Z”Ne using optical-model as defined in table 5.
parameters
set 4
fers of 0, 2, 4 and 0 respectively. A combination of L = 3 and L = 1 distributions was used in obtaining the fits to the 5.63f5.80 MeV states. Particular difficulty was encountered with the fits to the 1.63 MeV, 2+ level; no set of optical-model parameters was found that gave a good fit for this state. It was noted earlier in subsect. 3.2 that an absolute comparison of the DWBA
Fig. 6. DWBA
’
I
I
I
1 I
Qcm
01’
0.0 Mel’
I
fits to the
I
measured
1
angular
distributions for was used in obtaining
I
. .
I
I
I
MeV
I
.
l
%m
I
. . .
I 70
.
42 4 MeV
269
4;
425MeV
I I I 30 50
,
,
I
-
-
!-
I
IO
i:?,,
IO
100
100
.
30
.
50
l.
.*
I
9
42 4
35
26
+I;
l
. 70
.
MeV
MeV
MeV
MeV
1
II
10
I
I
30
I
50
8 cm
I
269
II
70
Me?/
MeV
’
----02
672 +
of L = 3 and L = 1 distributions
I---3;
563+580
the “Ne(p , t)“‘Ne reaction. A combination the fits to the 5.63 + 5.80 MeV states.
I
lt..
I IO
i?_ I-
IO
100 -
lo:-
IOO-
I
2*NeCp,
171
t)ZoNe
predictions with experiment cannot be made because of current ~n~rtainties in the theory of two-nucleon transfer reactions. Comparison of the relative magnitudes of the DWBA cross sections with experiment can, however, be made. The manner in which the calculated angular distributions were normalized to the experimental results is best illustrated with reference to fig. 6. For the L = 0 transfers the peak amplitudes were matched at some well-defined maximum (as shown); in the other cases a leastsquares fit of the theoretical curve to the experimental points was made. While exception may be taken to this procedure in normalization, the results are, nevertheless, internally consistent as far as the comparison between different sets of spectroscopic amplitudes is concerned, because of the relative insensitivity of the shapes of the angular distributors to these amplitudes. From the DWBA fits (as shown in fig 6) normalization factors, defined as CT,,J ~~~~~~ were determined. These factors should, of course, have the same value for TABLE
Normalization
6
factors, defined as b&q,bWBA, for the 2ZNe(p, t)*ONe reaction at 42.4 MeV
State
Bound-state
parameters
Spectroscopic amplitudes
(see table 2)
Energy
Jn
A
B
C
D
E
F
G
( MeV) 0.00 6.72
1.63 4.25
01+ 02+ 21+ 41
+
0,’ 02 +
0.00
6.72 1.63 4.25 0.00
21+ +
6.72 1.63 4.25
01f 02 + 21+ 41 + 1,31-
0.00 6.72 1.65 4.25 5.80
41
1.39 1.45 0.94 0.23 rms dev. 0.49
1.17 1.44 1.00 0.38 0.39
1.07 I.52 1.03 CL38 0.41
0.76 2.12 0.76 0.36 0.66
0.63 2.49 0.62 0.2? 0.87
1.23 1.91 0.56 0,30 0.63
0.52 3.03 0.31 0.14 1.18
1.96 0.55 1.05 0.45 rms dev. 0.60
I .76 0.57 1.18 0.49 0.51
1.65 0.61 1.25 0.49 0.47
1.36 1.00 1.09 0.56 0.29
1.26 1.27 1.01 0.46 0.33
2.02 0.82 0.75 0.42 0.61
1.25 1.83 0.63 0.29 0.59
1.20 1.20 1.45 0.15 rms dev. 0.50
1.04 1.21 1.59 0.16 0.52
0.94 1.28 1.61 0.16 0.54
0.75 1.89 1.18 0.18 0.62
0.67 2.06 1.13 0.14 0.7E
1.23 1.68 0.96 0.14 0.56
0.61 2.72 0.59 0.082 1.01
0.80 0.86 1.79 0.55
0.72 0.90 1.80 0.59 9.4 0.24 0.47
0.62 0.96 1.79 0.64 10.1 0.21 0.48
0.46 1.53 1.38 0.63 3.7 0.14 0.46
0.88 1.33 1.20 0.59 5.1 0.43 0.29
5.63 rms dev. 0.47
1.39 0.61 2.33 0.12
Wildenthal
Kilo
Arima
Zuker 1.29 0.71 0.25
The average of the normalization factors within each group corresponding to a spectroscopic amplitudes and Exed bound-state parameters is set at unity for Excluded in this averaging are the 11- and 3I - states. The rms deviation from normalization factors is indicated for each group. Particulars on the bound-state are given in tables 2 and 3, and for the spectroscopic amplitudes in table 1.
given set of convenience. unity of the calculations
172
W. R. FALK et al.
each state if the reaction mechanism has been properly described and the correct nuclear structure information has entered the calculation. The degree to which these normalization factors are the same, then, reflects the extent to which the above criteria are satisfied. These normalization factors were determined from DWBA calculations involving the various sets of spectroscopic amplitudes and the different bound-state parameters. Since the absolute value of the normalization factor is not established, these factors have, for convenience, been averaged to unity within each group corresponding to a given set of spectroscopic amplitudes and fixed bound-state parameters. Table 6 gives the normalization factors for the “Ne(p, t)“Ne reaction at 42.4 MeV. Detailed studies of the effect of the bound-state parameters were made at this energy. The rms deviation from unity of the normalization factors has been calculated for each of the various cases as shown in the table. It is observed that calculations with bound-state parameters A, B and C, with a few exceptions, yield quite similar normalization factors. A variation of 7 MeV in the single-particle potentials thus yields variations in the extracted normalization factors of the order of 30 ‘A. Calculations TABLE7 Normalization
factors, defined as ~~~~/urnva~,for the 22Ne(p, t)zONe reaction using bound-state narameters B of table 2 State
Jr
-%I, WW
Wildenthal
Kuo
Arima
Zuker
26.9
I .08 0.37 1.41 0.88
1.31 0.14 1.50
1.04
0.88 0.34 2.20 0.36
0.50 0.22 2.87 1.10 3.2 1.38
Energy (MeV)
o,+
0.00
21+ 41+ 1131-
6.72 1.63 4.25 5.80 5.63
01+ 02 + 21 + 41 + 1131-
0.00 6.72 1.63 4.25 5.80 5.63
35.1
1.32 0.93 1.40 0.43
1.80 0.34 1.52 0.50
1.14 0.80 2.12 0.17
0.66 0.51 2.22 0.53 4.0 0.47
01+ 02 + 21+ 41 + 1131-
0.00 6.72 1.63 4.25 5.80 5.63
42.4
1.22 1.51 1.05 0.40
1.69 0.55 1.14 0.47
1.03 1.20 1.57 0.16
0.39
0.55
0.67
0.61 0.76 1.52 0.50 8.0 0.20 0.77
02
+
rms dev.
The average of the normalization factors in each of the columns corresponding to a given set of spectroscopic amplitudes is set at unity for convenience. Excluded in this averaging are the I- and 3 - states. The rms deviation from unity of the normalization factors in each column is indicated.
*‘Ne(p,
TABLE
Normalization
factors,
Elab (MeV)
Jn
Energy (MeV)
01+ 02+ 21+ 41 +
0.00
01+ 02+ 21+ + 41 II31+ 01 + 02 21 + + 41 II31-
8
for the 22Ne(p, t)20Ne reaction defined as uCexs/uDWaA, parameters E of table 2
State
II31-
173
t)‘ONe
Wildenthal
using bound-state
Kuo
Arima
Zuker
26.9
0.57 1.40 1.05 0.83
0.90 0.65 1.53 1.27
0.53 1.28 1.80 0.41
0.32 0.84 2.08 1.58 1.47 0.36
0.00 6.12 1.63 4.25 5.80 5.63
35.1
0.67 2.16 0.90 0.33
1.20 1.32 0.50
0.68 1.83 1.53 0.16
0.42 1.19 1.65 0.62 1.50 0.19
0.00 6.12 1.63 4.25 5.80 5.63
42.4
0.64 2.54 0.63 0.21
1.15 1.15 0.92 0.42
0.63 1.95 1.07 0.13
rms dev.
0.68
0.32
0.64
0.38 1.26 1.14 0.52 3.08 0.11 0.55
6.12 1.63 4.25 5.80 5.63
1.oo
The average of the normalization factors in each of the columns corresponding to a given set of spectroscopic amplitudes is set at unity for convenience. Excluded in this averaging are the land 3- states. The rms deviation from unity of the normalization factors in each column is indicated.
with bound-state parameters D and E are also found to be closely similar, a result that is not too surprising in view of the fact that the differences in these parameters are relatively small where only the ld+-2s, orbits are involved. Finally, calculations with the bound-state parameters F and G, both of which are significantly different from the other parameters employed, yield normalization factors which, also, are significantly different from the others. For an overall comparison of the extracted nuclear structure information at the various proton bombarding energies tables 7 and 8 give the normalization factors at the three energies using bound-state parameters B and E respectively. In the former table the spectroscopic amplitudes from the wave functions of Wildenthal provide the most consistent description; the twelve normalization factors show an rms deviation from unity of only 0.39. Less satisfying though, is the observation that a strong energy dependence of the normalization factors exists for the 0; and 4: states. Calculations with bound-state parameters E, as shown in table 8 yield the most satisfactory results for Kuo’s wave functions. Again a rather strong energy dependence of the normalization factors, particularly for the 4: state, is observed.
174
W. R. FALK et al.
4. Discussion
and conclusions
The study of the “Ne(p, t)20Ne reaction at proton bombarding energies of 26.9,3.5.1 and 42.4 MeV permits several important conclusions to be drawn. Firstly, the singleparticle binding energies (bound-state parameters) are found to have a marked effect on the normalization factors (table 6). A proper understanding of this aspect of the problem thus becomes essential before more reliable nuclear structure information can be extracted from two-nucleon transfer reactions. Secondly, the extracted normalization factors for certain states show a strong dependence on the proton bombarding energy (tables 7 and 8). Since this energy dependence varies from state to state it can not readily be ascribed to an incorrect choice of optical-model parameters. That is, a different, or more appropriate choice of optical-model parameters would not necessarily remedy this situation. Nevertheless, this phenomena may be due to an inadequate treatment of the reaction problem in the region where the dominant part of the reaction occurs, namely the nuclear surface. Within the limits imposed by the above uncertainties on the normalization factors one can conclude that both the wave functions of Wildenthal et al. ‘) and Kuo “) provide a good description of the low-lying positive parity states in 20Ne and of the 22Ne ground state. In view of the fact that the Id+ orbit was omitted from the calculations of the spectroscopic amplitudes using Kuo’s “) matrix elements, one might have expected poorer agreement in this case. However, it has very recently been demonstrated by McGrory 21) that, when the Id, orbit is omitted from the model space, and the Kuo matrix elements for the Id, and 2s+ orbits are used without alteration, the resulting calculated energy levels and wave functions are very similar to the results from a complete three-shell calculation. Thus the omission of the Id+ orbit from these calculations should not have compromised the significance of these results. The spectroscopic amplitudes from Zuker’s “) wave functions likewise also yield reasonably good normalization factors for the positive parity states; the negative parity states are not as well described. Since the determination of the Arima “) matrix elements employed only those nuclei with two to four nucleons outside of the I60 core, the applicability of these matrix elements to the 22Ne ground state may justifiably be questioned. The cross section for the transition to the 4+, 4.25 MeV state is particularly poorly predicted in most cases with the use of the latter spectroscopic amplitudes. The excitation of the 2- state at 4.97 MeV is of particular interest since this transition is forbidden on the basis of a one-step direct pickup process. Several plausible explanations may be advanced for the excitation of this state. First, inelastic processes in the incoming or outgoing channel would permit transitions to the 2- state. The role of inelastic processes in two-nucleon transfer reactions has been discussed recently by Ascuitto and Glendenning “). A coupled-channels analysis of this reaction according to their prescription might prove useful in exploring such two-step processes
22Ne(p,
175
t)20Ne
TABLE 9 Differential
cross sections
for the reaction
13.5 16.8 19.6 22.4 25.7 28.0 33.5 36.3 39.0 44.5 50.0 55.4 60.8 66.1 71.4 76.6
~c.m.
t)*ONe*
(4.97 MeV, 2-) Ep = 42.4 MeV
Ep = 35.1 MeV
Es = 26.9 MeV
e c.m.
**Ne(p,
error
Olb/sr)
Cubisr)
7.3 10.0 11.7 11.2 14.7 14.2 13.7 14.2 13.0 10.9 11.3 12.4 7.9 6.4 6.7 7.4
1.5 1.2 1.1 1.4 1.3 1.6 1.5 1.7 1.3 1.1 1.1 1.1 1.0 0.7 1.0 1 .o
e C.rn.
~c.m. (pbisr)
22.2 27.6 33.1 38.6 47.3 54.8 60.2
12.9 15.0 8.5 6.0 3.8 3.1 4.7
error
eE.rn.
;;;;;,
2.5 1.6 1.1 0.9 0.9 0.7 0.8
error Ocbisr)
Ocb/sr) 13.2 16.5 19.3 22.0 25.3 27.5 33.0 38.4 43.8 49.2 54.6 59.9 65.2 70.4
14.9 9.6 13.4 9.8 8.7 9.6 5.0 2.2 1.2 3.2 2.2 2.0 0.9 0.9
2.8 1.2 1.7 1.0 1.2 1.1 0.6 0.5 0.5 0.6 0.4 0.6 0.3 0.3
in transfer reactions. Secondly, it is possible that the 6-9 ‘4 D-state admixture in the triton, which is always neglected in DWBA calculations, is of some importance in two-nucleon transfer reactions. The transfer in this case would then proceed by the pickup of a neutron pair in the 3P state. Cross sections for this state at each of the three proton bombarding energies are given in table 9. The authors want to express their appreciation to Dr. I. S. Towner, Oxford, who calculated the two-nucleon spectroscopic amplitudes for us. Dr. J. M. Nelson provided much valuable assistance in the use of his DWBA computer program, without which, these calculations could not have been carried out. Numerous fruitful discussions with Dr. L. R. Scherk regarding optical-model potentials were much appreciated.
References W. R. Falk and P. KuliFic, Bull. Am. Phys. Sot. 15 (1970) 766 1) A. McDonald, and M. A. Clark, Nucl. Phys. 41 (1963) 448 2) H. E. Gove, A. E. Litherland 3) 4) 5) 6) 7)
E. Almqvist and J. A. Kuehner, Nucl. Phys. 55 (1964) 145 J. D. Pearson, E. Almqvist and J. A. Kuehner, Can. J. Phys. 42 (1964) 489 C. Lederer, J. Hollander and I. Perlman, Table of isotopes (Wiley, New York, 1967) A. Arima, S. Cohen, R. D. Lawson and M. H. MacFarlane, Nucl. Phys. A108 (1968) 94 B. H. Wildenthal, J. B. McGrory, E. C. Halbert and P. W. M. Glaudemans, Phys. Lett. 26B (1968) 692
176
W. R. FALK et al.
8) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71
9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
A. P. Zuker, Phys. Rev. Lett. 23 (1969) 983 W. R. Falk, R. J. Kidney, P. Kulisic and G. K. Tandon, Nucl. Phys. A157 (1970) 241 I. S. Towner and J. C. Hardy, Adv. in Phys. 18 (1969) 401 I. S. Towner, private communication I. S. Towner and W. G. Davies, Oxford Nuclear Physics lab. report 27/69 (and Atlas Program Library Report No. 16) (1969) J. M. Nelson and B. E. F. Macefield, Oxford Nuclear Physics lab. report 18/69 (and Atlas Program Library Report No. 17) (1969) N. K. Glendenning, Phys. Rev. 137 (1965) B102 M. Moshinsky, Nucl. Phys. 13 (1959) 104 J. M. Nelson, N. S. Chant and P. S. Fisher, Nucl. Phys. Al56 (1970) 406 V. S. Mathur and J. R. Rook, Nucl. Phys. A91 (1967) 305 R. L. Jaffe and W. J. Gerace, Nucl. Phys. Al25 (1969) 1 N. Nakanishi, Y. Chiba, Y. Awaya and K. Matsuda, Nucl. Phys. A140 (1970) 417 J. B. McGrory, Phys. Lett. 33B (1970) 327 R. J. Ascuitto and N. K. Glendenning, Phys. Rev. C2 (1970) 1260
2-F
I
Nuclear Physics A167 (1971) 157-176; Not to be reproduced
by photoprint
@ North-Holland
Publishing
or microfilm without written permission
THE 22Ne(p, t)“Ne
Co., Amsterdam from the publisher
REACTION
AT 26.9, 35.1 AND 42.4 MeV W. R. FALK, P. KULISIC + and A. MCDONALD University
of Manitoba,
Winnipeg,
Received 30 December
Canada ++
1970
Abstract: Angular distributions of tritons from the 2ZNe(p, t)20Ne reaction have been measured at proton bombarding energies of 26.9, 35.1 and 42.4 MeV. All the known energy states up to an excitation energy of 6.72 MeV in “‘Ne were excited, including the unnatural parity 2- level at 4.97 MeV. DWBA calculations have been made using the finite-range two-nucleon transfer code of Nelson and Macefield. Various wave functions describing Z”Ne and 22Ne, taken from the work of Arima et al., Wildenthal et al., Kuo and Zuker, were used in calculating the spectroscopic amplitudes for transitions to the various final states in 20Ne. Comparison of the nuclear structure information extracted at the different bombarding energies is emphasized. The sensitivity of the DWBA results to the binding energy of the singleparticle states is discussed. E
NUCLEAR
REACTIONS
“Ne(p,
t), E = 26.9, 35.1, 42.4 MeV; measured
o(e),
1. Introduction
Most studies of the two-nucleon (p, t) transfer reaction have been performed at only one value of the proton bombarding energy. The extraction of nuclear structure information from such studies involves numerous uncertainties, one of which is the energy dependence of the kinematical aspects of the reaction. In an attempt to check the reliability with which the nuclear structure information can be extracted in two-nucleon transfer reactions, the 22Ne(p, t)20Ne reaction was investigated ‘) at proton bombarding energies of 26.9, 35.1 and 42.4 MeV. The level structure for low-lying states in “Ne is well known 2- “) and h ence only transitions to levels of known spin and parity were involved. Several descriptions of nuclei in the sd shell within the restricted shell-model space of the ld,--2s, orbits have recently been presented 6*‘). Such descriptions are found to predict satisfactorily for example, the data on spectra, transition rates and the static multipole moments. In both the work of Arima et al. “) and Wildenthal et al. ‘) the two-body matrix elements were determined by adjusting them to minimize the rms deviation between the calculated and experimentally observed energy levels. Only the energy levels of nuclei with two to four nucleons outside the 7 Present address: Institute R. BoSkovid, Zagreb, Yugoslavia. tt Supported in part by the Atomic Energy Control Board of Canada. 157
158
W. R. FALK el al.
’ 6O core were considered in deducing the matrix elements in the former investigation, whereas in the latter, energy levels of nuclei with A > 20 and N, 2 < 14 were taken into account. More elaborate shell-model descriptions of nuclei in this mass region have been presented by Kuo “) and Zuker “). The latter considers the Ip, as well as the Id+ and 2s+ orbits, although the Id, orbit is omitted. Within this framework a description of the negative parity states is thus also provided. Kuo’s “) shell-model calculation encompasses the entire sd shell. However, in calculating the spectroscopic information for the DWBA calculations it was necessary to truncate his results omitting the Id, orbit, since the shell-model computer program was restricted to handling only two shell-model orbitals. The foregoing shell-model descriptions of the nuclei “Ne and 22Ne were utilized in determining spectroscopic amplitudes for the DWBA calculations for the 22Ne(p, t) 20Ne reaction.
2. Experimental The experimental has been described
procedure and results
layout of the University in an earlier publication
of Manitoba
sector focused
cyclotron
’ “). The overall experimental arrangement and data-taking procedure was the same as that used for the 20Ne(p, t)18Ne reaction lo) except that 3He and cr-particles were not recorded in the present experiment. Neon gas (obtained from Monsanto Research Corp.), enriched to 99.6 % 22Ne was contained in a 6 cm diameter gas cell at a pressure of about one atmosphere. The pressure in the gas cell was monitored by observing the elastically scattered protons in two NaI scintillation counters placed at 373” on either side of the proton beam. Entrance and exit windows from the gas cell consisted of 25.4 pm Kapton-H foil (manufactured by Du-Pont) which resulted in negligible energy straggling of the outgoing tritons. Triton spectra recorded at 26.9 and 42.4 MeV proton bombarding energy are shown in fig. 1. All the low-lying states -O.O(O+), 1.63(2+), 4.25(4+), 4.97(2-), 5.63(3-)+ 5.80(1-) and 6.72(0+) MeV states-are clearly observed in the spectra. In addition, numerous more highly excited states, particularly at 42.4 MeV bombarding energy, are visible in the spectra. The overall energy resolution obtained varied between 140 and 170 keV, the largest contribution of which was due to the energy spread of the incident proton beam. Beam currents between 10 and 80 nA, depending on the angle of measurement, were used in recording the data. Angular distributions for the observed states are shown in fig. 2 at the three measured energies. Both the L = 0 (0.0and 6.72 MeV states) and L = 2 (1.63 MeV state) angular distributions retain their characteristic shapes despite the variation in outgoing triton energies from approximately 11 to 33 MeV (ground state Q-value equals -8.664 MeV). The other angular distributions show greater variations in
159
22Ne(p, t)‘ONe
shape particularly at the lowest energy of 26.9 MeV. Efforts were made to ensure that the errors in the relative cross sections for the different energies were minimized. Toward this end the same detector geometry was used throughout and comparative 15022
EP
Ne(p,tj2’Ne =269
MeV
‘1 AB = 35”
CHAN
N EL
~0.
22Ne (p,t)20Ne Ep = 42 4 MeV
CHANNEL
Fig. 1. Triton
spectra
recorded
0
600
500
400
NO
at 26.9 and 42.4 MeV proton
bombarding
energy.
measurements at 42.4 MeV at the beginning and end of this series of experiments were made. The results from these latter measurements were the same within the statistical error limits. Error bars shown in fig. 2 represent the statistical counting
160
W. R. FALK ei al.
IQ00
l
26.9
NIV
.
33.1
MCV
D
42.4
NIV
800
200
40
0
0
20
40
60
80
0’
1 I
0
,
20
1
I
40 8 c-m.
I
,
!
60
Fig. 2. Angular distributions of tritons for statajs in *#Ne from the 2ZNe(p, t)2W8 raaction. solid Iines arc smooth lines drawa thrcrugh the data points.
I
80
The
161
22Ne(p, t)‘ONe 240-
22
NCIP.l P [email protected]
1
“Ne
(P.t?“N~(672MeV,0+I
and , 80 MeV.I- 1 026.9
200-
Ye”
n 35.1
MIV
35.1
M.”
0 42.4
Yev
0 42.4
Ye”
40-
160-
z
McV
.26.9 .
< 0 5
120-
q “j
30-
2: I
414 60’
20-
40-
10
0;
I
0
10
I
20
30
40
50
60
70
Of
00
I 0
I
IO
0 c.m.
r 20
I
30
I
I
I
40
50
60
I
70
I
80
e cm
Fig. 2 (continued).
errors only; surements.
an overall
normalization
uncertainty
of + 10 % applies
to the mea-
3. DWBA calculations 3.1. DWBA FORMALISM
A summary of the DWBA formalism for two-nucleon transfer reactions was presented in an earlier paper lo). For completeness, this summary is repeated here. Following the treatment given by Towner and Hardy 11) we write for the DWBA expression
for the differential
cross section
A(a, b)B:
do - = .&!!C 5 dQ (27~rh~)~ k,
x
c
[“111j~‘[nzz*i23
S:&i,
1, jl][n2
I,
j,];
JT)
where [nr Z,j,] and [n,Z,j,] are the quantum numbers defining the single particle states of the transferred nucleons, and L, S, J and Tare the quantum numbers of the transferred pair taken as a cluster. All the nuclear structure information is contained in the factor Sin, called the spectroscopic amplitude, D(S, T) is a quantity depending on the spin and isospin dependence of the nucleon-nucleon interaction and b&- is essentially a spectroscopic factor for the light particles. The amplitudes Bi
162
W. R. FALK
et al.
describe the kinematical aspects of the reaction. It is clear from the sum over [nl Z,j,] and [Q I2j,] appearing inside the brackets that different configurations for the two nucleons can contribute and these are added coherently. The nuclear structure information can thus not be factored from the nuclear reaction dependence in twonucleon transfer reactions as in the case of single-nucleon transfer reactions. In the expression (3.1) spin-orbit forces have been neglected. The spectroscopic amplitude Sk, is a measure of the overlap of the wave functions of the target nucleus and the final nucleus plus two nucleons defined by quantum numbers [n, I, j,] and [a2Zzj,]. Formally it is defined as
x IcI$$+“&~, ~1, rz) d& du, dr, .
(3.2)
Here tB refers to the coordinates of the A -2 nucleons in nucleus B, and the square brackets within the integral sign denote vector coupling. Given a certain model, say the shell model, for which the wave functions can be determined for the nuclei A and B, the spectroscopic amplitudes can be calculated. Such calculations were carried out for the “Ne(p, t)“Ne reaction for different wave functions describing 22Ne and “Ne as deduced by various investigators. These were described in some detail in the introduction. The actual calculation of the spectroscopic amplitudes was performed by Towner 12) using the shell-model program of Towner and Davies 13). Results of these calculations are shown in
Two-nucleon
State
Exp. ext. energy
spectroscopic
02 +
0.00
6.12
Arima (lP# (ld;)’ (2s+Y
-0.866 -0.168
(lP+Y (Id+)’ (2s$
-0.050 -0.075
11-
5.80
(lP+2@
21 +
1.63
(ld# (ldz2s+)
31-
5.63
(lp+ldz)
41 +
4.25
reaction
Configuration
(MeV)
01+
TABLE 1 amplitudes for the ZZNe(p, t)‘ONe
(Id*)’
“)
Wildenthal
-0.943 -0.282
“)
Kuo b,
-0.859 -0.304
Zuker
‘)
0.246 1.021 0.206 0.020 0.062 0.083
0.001 0.118
0.017 0.214
-0.314 0.094
0.328 -0.224
-0.212 0.297
-0.704 -0.090
- 1.279
- 1.084
-1.130
-0.651
-0.231
1.212
“) Relative signs of the sd components of the wave functions have been changed (see text for explanation). b, The Id+ orbit has been omitted from Kuo’s results prior to calculation of the spectroscopic amplitudes. ‘) Calculations limited to configurations of lowest seniority, i.e. v S 2.
22Ne@, t)*‘Ne
163
table 1 for various final states in 2oNe. As noted ea rli er , excep t for the wave functions of Zuker “) only the Id+-2s, orbits were considered; the Id, orbit was omitted from the model space prior to calculation of the spectroscopic amplitudes using Kuo’s “) matrix elements. A point of uncertainty arises in regards to the spectroscopic amplitudes obtained from the wave functions of Arima et al. “) and Wildenthal et al. ‘). In both cases the two-body matrix elements, determined from a fit to the experimental energy levels of sd shell nuclei, leave the relative signs of the sd components of the wave function undetermined. For this reason calculations were also carried out with the relative signs between the s and d components changed. It was found that only with this sign change could reasonable results be obtained. 3.2. DWBA
COMPUTER
PROGRAM
The DWBA two-nucleon transfer code used in the present calculations was that of Nelson and Macefield 14). While no attempt will be made to describe the various features of this code in detail, several pertinent aspects should be mentioned. A finite range option is incorporated in this code that allows the Gaussian parameters of the interaction range and the triton size to have different values. For all the calculations employing the finite range option an interaction range of 1.6 fm was adopted. The triton size parameter was fixed at 0.240 fm-‘. Several methods of calculating the wave function of the transferred two-nucleon pair have been presented 11,15). In the Glendenning theory 15) the two transferred nucleons are described by harmonic oscillator wave functions for which a separation into functions of the relative motion and the motion of the center of mass of the pair can readily be performed using the Moshinsky transformation 16). The incorrect asymptotic form of the harmonic oscillator wave function for the c.m. motion is replaced by a Hankel function. This calculation assumed a zero interaction range but takes into account the finite size of the triton by using a Gaussian wave function for this particle. An alternative to the above procedure allowing much greater flexibility is to describe the transferred particles by a product of Saxon-Woods wave functions with different radial arguments. The binding energy of each nucleon can then be independently varied. DWBA predictions employing the different methods of calculating the wave function of the transferred pair will be presented and discussed in subsect. 3.3. Further information regarding the details of these calculations can be found in refs. 14*17). The magnitudes of the cross sections predicted by eq. (3.1) are found to be very sensitive to a number of factors entering the DWBA calculation, chief among which are the single-particle binding energies (subsect. 3.3) and the optical-model parameters (subsect. 3.4). Several workers ‘*, ’ “) have investigated this problem but to date there is no adequate description of the reaction that permits reliable predictions of the absolute magnitudes of the cross sections. Consequently, only the relative magnitudes of the DWBA cross sections will be considered in the comparison with the experimental results.
164
W. R. FALK
3.3. SINGLE-PARTICLE
BINDING
et al.
ENERGIES
In the analysis of the “Ne(p, t)“Ne reaction lo) it was noted that variations of the order of 40 % in the predicted cross sections were obtained depending on the bound-state parameters employed. Current theories of two-nucleon transfer reactions do not provide an adequate prescription for calculating the bound state wave functions of the transferred nucleons. Recently Jaffe and Gerace lg) have presented arguments (applicable to ground state transitions of doubly even nuclei with closed neutron and proton shells) which show that the appropriate neutron binding energy to use, for example, in the case of the laO(p, t)160 g.s. reaction, is that of the neutron in “0. This adequately accounts for the pairing energy of the two neutrons. TABLE 2 Summary
of the different
bound-state
Range
parameters Singleparticle well depth
used in the DWBA
calculations
Comments
(MeV) A B C D E F G
“) Oscillator
finite finite finite finite finite finite zero
parameter
6.0 5.5 6.0 5.5 6.0 5.5
55.0 59.0 62.0
59.0
intermediate case of binding (see table 3) binding energy equally divided R = R,,,. = 1.3 fm, A = A,.,. = 0.7gfm Glendenning-type calculation “)
Y = 0.31 frnm2.
In the more general case of nuclei removed from closed shells the situation is much less clear. For this reason a number of calculations were carried out in which the single-particle binding energy was varied. A summary of the particulars of these calculations is presented in table 2. The calculations labelled A, B and C correspond to cases in which the parameters of the potential well were held fixed for calculating lp,, Id, and 2s, single-particle energies. A Saxon-Woods form for the potential was assumed with a radius parameter of 1.20 fm and a diffuseness of 0.70 fm. The spin-orbit potential, V,,,, , varied between 5.5 and 6.0 MeV. The binding energies so generated are shown in fig. 3 for the case of V,,,, = 5.5 MeV. For the well depth of 55 MeV the mean of the d;-s+ neutron separation energies corresponds to the Separation energy of a neutron from ‘lNe . At a potential depth of 62 MeV the mean of the d;-s, neutron separation energy corresponds roughly to one half of the twoneutron separation energy from 22Ne (for the 1.63-4.25 MeV transitions in “Ne). The arrows in the lower portion of fig. 3 correspond to one half of the two-neutron separation energy from ‘*Ne for the 0.0, 1.63,4.25 and 6.72 MeV final states in 20Ne.
22Ne(p,
165
t)*ONe
Calculation E in table 2 represents the opposite extreme of the calculations A, B and C in that the experimental two-neutron separation energy was simply divided equally between the two neutrons, regardless of the orbit from which the neutron was removed. A zero-range calculation, according to the Glendenning formalism, is represented by calculation G. Finally, a choice of single-particle binding energies, intermediate between the extremes represented above, is given by calculation D. Details for the binding energies for the different states for this calculation are given in table 3; essentially, the experimental two-neutron separation energy was divided among the single-particle states taking cognizance of the spin-orbit separation between the two orbits. One further choice of bound state parameters is indicated by
Y 0 40Z v, 2
I
I
I
4
6
8
0.0 1.63 4.25 I
BINDING
10
6.72 I
I
I
I
12
14
ib
18
ENERGY
I
20
22
(MeV)
Fig. 3. Single-particle well depth as a function of the binding energy A spin-orbit potential of 5.5 MeV and radius and diffuseness parameters respectively, were used in the calculations. The positions of the arrows the figure indicate one half of the two-neutron separation energy 1.63, 4.25 and 6.72 MeV final states in ‘ONe.
for a neutron in ‘*Ne. of 1.20 fm and 0.70 fm in the lower portion of in **Ne for the 0.0,
TABLE 3 Representative
single-particle
State J”
02 + 11-
The O+ states
(MeV)
Experimental two-neutron separation energy (MeV)
0.00 6.72 5.80
17.15 23.87 22.95
Energy
01+
binding
are not calculated
for Zuker’s
energies
for calculation Binding
1pt.
D of table 2 energy ld%
2s.t
9.43 13.13
7.72 10.74 4.87
18.08
9, wave functions
(MeV)
in this case.
166
W. R. FALK
et al.
TABLE 4
DWBA
cross-section
predictions
for the different
bound-state
parameters
as defined
in table
2
%xvBA B
A
C
D
E
26.9
01 + 41 +
L=O L=4
(30”) (74”)
40.6 1.36
42.3 0.93
35.1
01 + 41+
L=O L=4
(28”) (60”)
24.3 3.40
26.3 2.58
42.4
0,+ 41 +
L=O L=4
(26”) (50”)
21.8 7.47
15.1 4.73
The overall normalization is arbitrary but the relative cross Spectroscopic amplitudes from the wave functions of Wildenthal are given at the peak in the vicinity of the angle indicated.
11.6 3.39
14.7 3.10
15.9 3.80
F
13.5 4.17
sections are correctly indicated. were used. Cross-section values
calculation F in which the geometric parameters of the Saxon-Woods potential were changed to 1.30 and 0.78 fm (for the radius and diffuseness parameters respectively). An indication of the sensitivity of the magnitudes of the DWBA cross sections for the different calculations A-F described in table 2 is presented in table 4. Spectroscopic amplitudes from the wave functions of Wildenthal (see table 1) were employed in obtaining these results. Variations of the order of a factor of 2 are observed as the binding energy of the nucleons is varied. These variations are much less than those observed in the corresponding “Ne(p, t)“Ne calculations lo) where a large imaginary radius parameter of 1.9 fm was employed (see subsect. 3.4). The ratios of the calculated cross sections, while exhibiting much smaller variations than the factor of two noted above, do, however, still indicate that the choice of the correct bound state parameters is an important aspect of the DWBA calculations. Further discussion of this point will be presented later. 3.4. OPTICAL-MODEL
PARAMETERS
The form of the optical-model
potential
used in the DWBA calculation
is defined by
where j and Vc is the No spin-orbit At the time for the elastic deduced from
(r,,
a,) = [I+exp
{+]I-’
(3.3)
Coulomb potential due to a uniformly charged sphere of radius r&. term was employed. when these calculations were performed no optical-model parameters scattering of protons on “Ne were available. Hence the parameters the p+ “Ne elastic scattering lo) were adopted without modification
“Ne(p, for
these calculations.jThis
t)“Ne
set of optical-model
167
parameters
is shown in the first line
of table 5. For t + “Ne elastic scattering, likewise, there existed no experimental data. Numerous sets of optical-model parameters for the t + “Ne outgoing channel were tried in the DWBA calculations. Finally the set shown in the second line in table 5 was adopted for all the calculations on the basis of a best all around fit to the experimental angular distributions (see fig. 6). Although the energies in both the incoming and outgoing channels vary over a rather wide range (26.9 to 42.4 MeV (lab) for the protons, and approximately 11 to 33 MeV (lab) for the tritons), the same set of TABLE
Optical-model
parameters
used
p+ZZNe t fZoNe Set 1 Set 2 Set 3 Set 4
36.33 150.0 213.0 213.0 150.0 120.0
of the p+2ZNe Imaginary
Real volume
(MZ)
5
for the calculation
volume
and
t+ZoNe
distorted
Imaginary
(2)
(fZ)
(ME&
(f&
(fZ)
Wo (MeV)
1.197 1.54 1.45 1.40 1.50 1.25
0.746 0.66 0.70 0.70 0.60 0.65
11.31 45.0 45.0 45.0 45.0 50.0
1.196 1.54 1.45 1.40 1.50 2.00
0.786 0.66 0.70 0.70 0.60 0.65
0.18 0.0 0.0 0.0 0.0 27.0
waves
surface
(fm”, 1.196
2.00
0.786
0.55
The parameter sets l-3 for t+“‘Ne were used for the calculation of the angular distributions shown in fig. 4. Angular distributions using optical parameters set 4 are shown in fig. 5.
optical-model parameters was used throughout. This approach was adopted since the use of different parameters at the various energies would have introduced many new uncertainties into the interpretation of the results. Justification for such a procedure can be made in view of the general insensitivity of the DWBA calculations to the proton parameters, and the observation that the optical-model parameters for mass-three particles exhibit little energy dependence ““). Somewhat better fits to individual angular distributions were obtained by optimizing parameters for that particular case. Thus the sets of t + “Ne optical-model parameters labelled 1, 2 and 3 in table 5 were found to give improved fits for the cases shown in fig. 4. A markedly different set of optical-model parameters for t + “Ne (set 4 in table 5), with an imaginary radius parameter of 2.00 fm, was also tried. This set was similar in many ways to that used for the t + 18Ne analysis ’ “). However, with such a large imaginary radius parameter, total reaction cross sections are incorrectly determined and absolute values for the (p, t) cross sections are much smaller than reasonable estimates would lead one to expect. With the other sets of t + 20Ne parameters both the total reaction cross sections and the absolute values of the (p, t) cross sections
168
W. R. FALK
et al.
are acceptable. For comparison, DWBA angular distributions calculated with opticalmodel parameters set 4 are shown in fig. 5 for the O:, 0: and 4: states in 20Ne. While somewhat inferior fits are obtained than with the more acceptable t + 2oNe parameters (figs. 4 and 6), the differences are not as great as one might have expected. 3.5. DWBA
RESULTS
The (p, t) reaction on spin-zero target nuclei has the simplifying feature that only one value of the angular momentum transfer L enters into the calculation. Selection m-1
I
22Ne(p,t)‘oNe
42.4 MeV
t
I
I
,
“Ne(p,t)
I
I
2CNe
/
I
42.4 MeV
I 0.0 MeV
-I
1000 -
IOO-
4.25 4:
MeV
Set
2
Set
3
100 -
$ a lo-
1000 $ -0
-
100 -
loo-
-
. ..
l
.
.* .*
.
.
.
IO
IO-
;T:
I IO
Fig. 4. DWBA
I
I I I 30 50 0 cm
I
1 1
70
I IO
I
I 30
I 8
I 50
I
I
cm
fits for the Oi+ and 4,+ states in *ONe using various model parameters
I 70
sets of t+20Ne
optical-
as defined in table 5.
rules, on the basis of a one-step direct pickup mechanism, give the result Jf = L, and restrict the parity to q = (- 1)” = (- l)Jf. The shapes of the angular distributions are largely insensitive to different choices of the bound-state parameters and to the spectroscopic amplitudes.
169
22Ne(p,t)20Ne
DWBA fits to the measured angular distributions (excluding the 4.97 MeV, 2- level) are shown in fig. 6. All calculations for the curves shown were performed with the optical-model parameters shown in the first two lines of table 2, and the spectroscopic amplitudes from the wave functions of Wildenthal et al. ‘) (see table 1). The curves for the 0.0, 1.63, 4.25 and 6.72 MeV states correspond to angular momentum trans6.72 MeV
100
02+ T----J 1
j
~p.269MeV) 100
1
10’
Ep.351
\ .n
MeV]
t
10
10
I I I IO 30
I
8 cm
I 50
I
I 1 70
8 cm
Fig. 5. DWBA fits for the O,+, 02+ and 41f states in Z”Ne using optical-model as defined in table 5.
parameters
set 4
fers of 0, 2, 4 and 0 respectively. A combination of L = 3 and L = 1 distributions was used in obtaining the fits to the 5.63f5.80 MeV states. Particular difficulty was encountered with the fits to the 1.63 MeV, 2+ level; no set of optical-model parameters was found that gave a good fit for this state. It was noted earlier in subsect. 3.2 that an absolute comparison of the DWBA
Fig. 6. DWBA
’
I
I
I
1 I
Qcm
01’
0.0 Mel’
I
fits to the
I
measured
1
angular
distributions for was used in obtaining
I
. .
I
I
I
MeV
I
.
l
%m
I
. . .
I 70
.
42 4 MeV
269
4;
425MeV
I I I 30 50
,
,
I
-
-
!-
I
IO
i:?,,
IO
100
100
.
30
.
50
l.
.*
I
9
42 4
35
26
+I;
l
. 70
.
MeV
MeV
MeV
MeV
1
II
10
I
I
30
I
50
8 cm
I
269
II
70
Me?/
MeV
’
----02
672 +
of L = 3 and L = 1 distributions
I---3;
563+580
the “Ne(p , t)“‘Ne reaction. A combination the fits to the 5.63 + 5.80 MeV states.
I
lt..
I IO
i?_ I-
IO
100 -
lo:-
IOO-
I
2*NeCp,
171
t)ZoNe
predictions with experiment cannot be made because of current ~n~rtainties in the theory of two-nucleon transfer reactions. Comparison of the relative magnitudes of the DWBA cross sections with experiment can, however, be made. The manner in which the calculated angular distributions were normalized to the experimental results is best illustrated with reference to fig. 6. For the L = 0 transfers the peak amplitudes were matched at some well-defined maximum (as shown); in the other cases a leastsquares fit of the theoretical curve to the experimental points was made. While exception may be taken to this procedure in normalization, the results are, nevertheless, internally consistent as far as the comparison between different sets of spectroscopic amplitudes is concerned, because of the relative insensitivity of the shapes of the angular distributors to these amplitudes. From the DWBA fits (as shown in fig 6) normalization factors, defined as CT,,J ~~~~~~ were determined. These factors should, of course, have the same value for TABLE
Normalization
6
factors, defined as b&q,bWBA, for the 2ZNe(p, t)*ONe reaction at 42.4 MeV
State
Bound-state
parameters
Spectroscopic amplitudes
(see table 2)
Energy
Jn
A
B
C
D
E
F
G
( MeV) 0.00 6.72
1.63 4.25
01+ 02+ 21+ 41
+
0,’ 02 +
0.00
6.72 1.63 4.25 0.00
21+ +
6.72 1.63 4.25
01f 02 + 21+ 41 + 1,31-
0.00 6.72 1.65 4.25 5.80
41
1.39 1.45 0.94 0.23 rms dev. 0.49
1.17 1.44 1.00 0.38 0.39
1.07 I.52 1.03 CL38 0.41
0.76 2.12 0.76 0.36 0.66
0.63 2.49 0.62 0.2? 0.87
1.23 1.91 0.56 0,30 0.63
0.52 3.03 0.31 0.14 1.18
1.96 0.55 1.05 0.45 rms dev. 0.60
I .76 0.57 1.18 0.49 0.51
1.65 0.61 1.25 0.49 0.47
1.36 1.00 1.09 0.56 0.29
1.26 1.27 1.01 0.46 0.33
2.02 0.82 0.75 0.42 0.61
1.25 1.83 0.63 0.29 0.59
1.20 1.20 1.45 0.15 rms dev. 0.50
1.04 1.21 1.59 0.16 0.52
0.94 1.28 1.61 0.16 0.54
0.75 1.89 1.18 0.18 0.62
0.67 2.06 1.13 0.14 0.7E
1.23 1.68 0.96 0.14 0.56
0.61 2.72 0.59 0.082 1.01
0.80 0.86 1.79 0.55
0.72 0.90 1.80 0.59 9.4 0.24 0.47
0.62 0.96 1.79 0.64 10.1 0.21 0.48
0.46 1.53 1.38 0.63 3.7 0.14 0.46
0.88 1.33 1.20 0.59 5.1 0.43 0.29
5.63 rms dev. 0.47
1.39 0.61 2.33 0.12
Wildenthal
Kilo
Arima
Zuker 1.29 0.71 0.25
The average of the normalization factors within each group corresponding to a spectroscopic amplitudes and Exed bound-state parameters is set at unity for Excluded in this averaging are the 11- and 3I - states. The rms deviation from normalization factors is indicated for each group. Particulars on the bound-state are given in tables 2 and 3, and for the spectroscopic amplitudes in table 1.
given set of convenience. unity of the calculations
172
W. R. FALK et al.
each state if the reaction mechanism has been properly described and the correct nuclear structure information has entered the calculation. The degree to which these normalization factors are the same, then, reflects the extent to which the above criteria are satisfied. These normalization factors were determined from DWBA calculations involving the various sets of spectroscopic amplitudes and the different bound-state parameters. Since the absolute value of the normalization factor is not established, these factors have, for convenience, been averaged to unity within each group corresponding to a given set of spectroscopic amplitudes and fixed bound-state parameters. Table 6 gives the normalization factors for the “Ne(p, t)“Ne reaction at 42.4 MeV. Detailed studies of the effect of the bound-state parameters were made at this energy. The rms deviation from unity of the normalization factors has been calculated for each of the various cases as shown in the table. It is observed that calculations with bound-state parameters A, B and C, with a few exceptions, yield quite similar normalization factors. A variation of 7 MeV in the single-particle potentials thus yields variations in the extracted normalization factors of the order of 30 ‘A. Calculations TABLE7 Normalization
factors, defined as ~~~~/urnva~,for the 22Ne(p, t)zONe reaction using bound-state narameters B of table 2 State
Jr
-%I, WW
Wildenthal
Kuo
Arima
Zuker
26.9
I .08 0.37 1.41 0.88
1.31 0.14 1.50
1.04
0.88 0.34 2.20 0.36
0.50 0.22 2.87 1.10 3.2 1.38
Energy (MeV)
o,+
0.00
21+ 41+ 1131-
6.72 1.63 4.25 5.80 5.63
01+ 02 + 21 + 41 + 1131-
0.00 6.72 1.63 4.25 5.80 5.63
35.1
1.32 0.93 1.40 0.43
1.80 0.34 1.52 0.50
1.14 0.80 2.12 0.17
0.66 0.51 2.22 0.53 4.0 0.47
01+ 02 + 21+ 41 + 1131-
0.00 6.72 1.63 4.25 5.80 5.63
42.4
1.22 1.51 1.05 0.40
1.69 0.55 1.14 0.47
1.03 1.20 1.57 0.16
0.39
0.55
0.67
0.61 0.76 1.52 0.50 8.0 0.20 0.77
02
+
rms dev.
The average of the normalization factors in each of the columns corresponding to a given set of spectroscopic amplitudes is set at unity for convenience. Excluded in this averaging are the I- and 3 - states. The rms deviation from unity of the normalization factors in each column is indicated.
*‘Ne(p,
TABLE
Normalization
factors,
Elab (MeV)
Jn
Energy (MeV)
01+ 02+ 21+ 41 +
0.00
01+ 02+ 21+ + 41 II31+ 01 + 02 21 + + 41 II31-
8
for the 22Ne(p, t)20Ne reaction defined as uCexs/uDWaA, parameters E of table 2
State
II31-
173
t)‘ONe
Wildenthal
using bound-state
Kuo
Arima
Zuker
26.9
0.57 1.40 1.05 0.83
0.90 0.65 1.53 1.27
0.53 1.28 1.80 0.41
0.32 0.84 2.08 1.58 1.47 0.36
0.00 6.12 1.63 4.25 5.80 5.63
35.1
0.67 2.16 0.90 0.33
1.20 1.32 0.50
0.68 1.83 1.53 0.16
0.42 1.19 1.65 0.62 1.50 0.19
0.00 6.12 1.63 4.25 5.80 5.63
42.4
0.64 2.54 0.63 0.21
1.15 1.15 0.92 0.42
0.63 1.95 1.07 0.13
rms dev.
0.68
0.32
0.64
0.38 1.26 1.14 0.52 3.08 0.11 0.55
6.12 1.63 4.25 5.80 5.63
1.oo
The average of the normalization factors in each of the columns corresponding to a given set of spectroscopic amplitudes is set at unity for convenience. Excluded in this averaging are the land 3- states. The rms deviation from unity of the normalization factors in each column is indicated.
with bound-state parameters D and E are also found to be closely similar, a result that is not too surprising in view of the fact that the differences in these parameters are relatively small where only the ld+-2s, orbits are involved. Finally, calculations with the bound-state parameters F and G, both of which are significantly different from the other parameters employed, yield normalization factors which, also, are significantly different from the others. For an overall comparison of the extracted nuclear structure information at the various proton bombarding energies tables 7 and 8 give the normalization factors at the three energies using bound-state parameters B and E respectively. In the former table the spectroscopic amplitudes from the wave functions of Wildenthal provide the most consistent description; the twelve normalization factors show an rms deviation from unity of only 0.39. Less satisfying though, is the observation that a strong energy dependence of the normalization factors exists for the 0; and 4: states. Calculations with bound-state parameters E, as shown in table 8 yield the most satisfactory results for Kuo’s wave functions. Again a rather strong energy dependence of the normalization factors, particularly for the 4: state, is observed.
174
W. R. FALK et al.
4. Discussion
and conclusions
The study of the “Ne(p, t)20Ne reaction at proton bombarding energies of 26.9,3.5.1 and 42.4 MeV permits several important conclusions to be drawn. Firstly, the singleparticle binding energies (bound-state parameters) are found to have a marked effect on the normalization factors (table 6). A proper understanding of this aspect of the problem thus becomes essential before more reliable nuclear structure information can be extracted from two-nucleon transfer reactions. Secondly, the extracted normalization factors for certain states show a strong dependence on the proton bombarding energy (tables 7 and 8). Since this energy dependence varies from state to state it can not readily be ascribed to an incorrect choice of optical-model parameters. That is, a different, or more appropriate choice of optical-model parameters would not necessarily remedy this situation. Nevertheless, this phenomena may be due to an inadequate treatment of the reaction problem in the region where the dominant part of the reaction occurs, namely the nuclear surface. Within the limits imposed by the above uncertainties on the normalization factors one can conclude that both the wave functions of Wildenthal et al. ‘) and Kuo “) provide a good description of the low-lying positive parity states in 20Ne and of the 22Ne ground state. In view of the fact that the Id+ orbit was omitted from the calculations of the spectroscopic amplitudes using Kuo’s “) matrix elements, one might have expected poorer agreement in this case. However, it has very recently been demonstrated by McGrory 21) that, when the Id, orbit is omitted from the model space, and the Kuo matrix elements for the Id, and 2s+ orbits are used without alteration, the resulting calculated energy levels and wave functions are very similar to the results from a complete three-shell calculation. Thus the omission of the Id+ orbit from these calculations should not have compromised the significance of these results. The spectroscopic amplitudes from Zuker’s “) wave functions likewise also yield reasonably good normalization factors for the positive parity states; the negative parity states are not as well described. Since the determination of the Arima “) matrix elements employed only those nuclei with two to four nucleons outside of the I60 core, the applicability of these matrix elements to the 22Ne ground state may justifiably be questioned. The cross section for the transition to the 4+, 4.25 MeV state is particularly poorly predicted in most cases with the use of the latter spectroscopic amplitudes. The excitation of the 2- state at 4.97 MeV is of particular interest since this transition is forbidden on the basis of a one-step direct pickup process. Several plausible explanations may be advanced for the excitation of this state. First, inelastic processes in the incoming or outgoing channel would permit transitions to the 2- state. The role of inelastic processes in two-nucleon transfer reactions has been discussed recently by Ascuitto and Glendenning “). A coupled-channels analysis of this reaction according to their prescription might prove useful in exploring such two-step processes
22Ne(p,
175
t)20Ne
TABLE 9 Differential
cross sections
for the reaction
13.5 16.8 19.6 22.4 25.7 28.0 33.5 36.3 39.0 44.5 50.0 55.4 60.8 66.1 71.4 76.6
~c.m.
t)*ONe*
(4.97 MeV, 2-) Ep = 42.4 MeV
Ep = 35.1 MeV
Es = 26.9 MeV
e c.m.
**Ne(p,
error
Olb/sr)
Cubisr)
7.3 10.0 11.7 11.2 14.7 14.2 13.7 14.2 13.0 10.9 11.3 12.4 7.9 6.4 6.7 7.4
1.5 1.2 1.1 1.4 1.3 1.6 1.5 1.7 1.3 1.1 1.1 1.1 1.0 0.7 1.0 1 .o
e C.rn.
~c.m. (pbisr)
22.2 27.6 33.1 38.6 47.3 54.8 60.2
12.9 15.0 8.5 6.0 3.8 3.1 4.7
error
eE.rn.
;;;;;,
2.5 1.6 1.1 0.9 0.9 0.7 0.8
error Ocbisr)
Ocb/sr) 13.2 16.5 19.3 22.0 25.3 27.5 33.0 38.4 43.8 49.2 54.6 59.9 65.2 70.4
14.9 9.6 13.4 9.8 8.7 9.6 5.0 2.2 1.2 3.2 2.2 2.0 0.9 0.9
2.8 1.2 1.7 1.0 1.2 1.1 0.6 0.5 0.5 0.6 0.4 0.6 0.3 0.3
in transfer reactions. Secondly, it is possible that the 6-9 ‘4 D-state admixture in the triton, which is always neglected in DWBA calculations, is of some importance in two-nucleon transfer reactions. The transfer in this case would then proceed by the pickup of a neutron pair in the 3P state. Cross sections for this state at each of the three proton bombarding energies are given in table 9. The authors want to express their appreciation to Dr. I. S. Towner, Oxford, who calculated the two-nucleon spectroscopic amplitudes for us. Dr. J. M. Nelson provided much valuable assistance in the use of his DWBA computer program, without which, these calculations could not have been carried out. Numerous fruitful discussions with Dr. L. R. Scherk regarding optical-model potentials were much appreciated.
References W. R. Falk and P. KuliFic, Bull. Am. Phys. Sot. 15 (1970) 766 1) A. McDonald, and M. A. Clark, Nucl. Phys. 41 (1963) 448 2) H. E. Gove, A. E. Litherland 3) 4) 5) 6) 7)
E. Almqvist and J. A. Kuehner, Nucl. Phys. 55 (1964) 145 J. D. Pearson, E. Almqvist and J. A. Kuehner, Can. J. Phys. 42 (1964) 489 C. Lederer, J. Hollander and I. Perlman, Table of isotopes (Wiley, New York, 1967) A. Arima, S. Cohen, R. D. Lawson and M. H. MacFarlane, Nucl. Phys. A108 (1968) 94 B. H. Wildenthal, J. B. McGrory, E. C. Halbert and P. W. M. Glaudemans, Phys. Lett. 26B (1968) 692
176
W. R. FALK et al.
8) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71
9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
A. P. Zuker, Phys. Rev. Lett. 23 (1969) 983 W. R. Falk, R. J. Kidney, P. Kulisic and G. K. Tandon, Nucl. Phys. A157 (1970) 241 I. S. Towner and J. C. Hardy, Adv. in Phys. 18 (1969) 401 I. S. Towner, private communication I. S. Towner and W. G. Davies, Oxford Nuclear Physics lab. report 27/69 (and Atlas Program Library Report No. 16) (1969) J. M. Nelson and B. E. F. Macefield, Oxford Nuclear Physics lab. report 18/69 (and Atlas Program Library Report No. 17) (1969) N. K. Glendenning, Phys. Rev. 137 (1965) B102 M. Moshinsky, Nucl. Phys. 13 (1959) 104 J. M. Nelson, N. S. Chant and P. S. Fisher, Nucl. Phys. Al56 (1970) 406 V. S. Mathur and J. R. Rook, Nucl. Phys. A91 (1967) 305 R. L. Jaffe and W. J. Gerace, Nucl. Phys. Al25 (1969) 1 N. Nakanishi, Y. Chiba, Y. Awaya and K. Matsuda, Nucl. Phys. A140 (1970) 417 J. B. McGrory, Phys. Lett. 33B (1970) 327 R. J. Ascuitto and N. K. Glendenning, Phys. Rev. C2 (1970) 1260