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Elastic neutron transfer in the scattering of Si isotopes
I
I
2.N
Nuclear
Physics
A234 (1974)
Not to be reproduced
167-l
by photoprint
84; @ North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
ELASTIC NEUTRON TRANSFER IN THE SCATTERING OF Si ISOTOPES K. D. HILDENBRAND, R. BOCK t, H. G. BOHLEN, P. BRAUN-MUNZINGER, D. FICK, C. K. GELBKE, URSULA WEISS and W. WEISS Max-Planck-Institut
.ftir Kerphysik, Received (Revised
6 May
Heidelberg,
Germany
1974
4 July 1974)
Angular distributions of the elastic scattering of “Si on 29Si and 3oSi have been measured for incident beam energies at E = 65 and 70 MeV with a time-of-flight spectrometer for heavy 3oSi(28Si, 29Si)z9Si was observed in addition to the ions. At 70 MeV the neutron transfer elastic channel. The pronounced oscillations in the elastic scattering distributions are interpreted as being due to an elastic transfer of neutrons between the colliding nuclei during the scattering process. This assumption is in accordance with some general features of the data and allows for the extraction of spectroscopic factors of the transferred neutrons.
Abstract:
E
NUCLEAR REACTIONS *9*30Si(28Si, “Si), MeV; measured n(0). 29* 3oSi levels deduced
E = 65, 70 MeV; 3oSi(28Si, z9Si), E = 70 spectroscopic factors. Enriched targets.
1. Introduction In the last few years there has been a growing interest in studying few nucleon transfer reactions induced by heavy ions. This interest has been stimulated by the fact that it has proved to be possible to deduce spectroscopic information (via DWBA codes which are suited for heavy ion reactions) comparable to that obtained by light ion induced reactions ‘). The dominant physical features of heavy ion collision are Coulomb interaction and strong absorption of the projectiles ‘). As a consequence, the angular distributions of the elastic scattering exhibit the well known Fresnel diffraction pattern 3), which can be well understood in the framework of models taking into account the strong absorption either by a corresponding optical potential or more directly by a smooth cut-off parametrization of the reflection coefficients. Such a description, however, does not provide any nuclear structure information. This information can be obtained from the deviations from the normal diffraction scattering. One of the effects which yield such deviations is the elastic transfer. This special kind of transfer reproduces the particles of the entrance channel and can be written as follows: A+ (A+x) -+ (A+x)+ A. The amplitude of such a transfer will contribute coherently to that of the elastic scattering, causing structures in the angular distribution, the strength of which depends on the height of the transfer amplitude. + Present address: Gesellschaft fiir Schwerionenforschung, Darmstadt, W.-Germany. 167
168
It is evident
K. D. HILDENBRAND
that the study of such a transfer
et al.
combines
several advantages:
it can
be studied by measuring the elastic scattering with normally large cross sections instead of bell-shaped transfer angular distributions. Therefore nuclear structure information such as spectroscopic factors can be obtained directly from the elastic scattering. An additional advantage is the fact that the transferred particles are bound in identical states in entrance and exit channel. Therefore only one spectroscopic factor influences the cross section contrary to other heavy ion transfer reactions, where the information is contained in a product of two spectroscopic amplitudes. Additionally, in heavy ion reactions at energies in the vicinity of the Coulomb barrier only the outer parts of the wave functions have to be known. This situation is similar to sub-Coulomb stripping reactions, which, however, have small cross sections in most cases ‘). During the last two years the elastic scattering of nuclei which differ only by a few nucleons has been investigated mainly with projectiles between 12C and 19F In all cases the shape of the measured angular distributions could be [refs. ‘-r’)]. reproduced very well by the assumption of an elastic transfer. For a quantitative description of this transfer, which is necessary for deriving spectroscopic information, two models can be applied: The treatment within the framework of DWBA theory “) and the LCNO model 12, 13). It was proved that both models are equivalent as long as the interaction which causes the transfer can be treated in first order *I). Throughout the interpretation of the present data the LCNO model has been applied because it was the only model available at the beginning of the analysis. In the experiments reported here the elastic scattering of isotopes, i.e. nuclei differing only by the neutron number was studied for the first time in the mass region above A = 20. The silicon isotopes A = 28, 29, 30 were chosen because the additional neutrons of the heavier isotopes are 2s, neutrons. The formalism of the elastic transfer models is much easier if the transferred nucleons are bound to the cores with I = 0. For spectroscopic reasons this experiment seemed to be interesting because the spectroscopic factors of the 2~ neutron in the 29Si ground state determined with (d, p) reactions ’ 4, 1 j) disagree as well as the theoretical predictions [refs. ’ 6, “)I. The results of the 28Si-29Si scattering have partly been described in a previous publication I’). Besides the elastic scattering of 28Si on 3 ‘Si at 70 MeV the neutron transfer channel transfer 3oSi(28Si, * 9Si)2 9Si was observed. By a DWBA analysis of this “nonelastic” only the product of the spectroscopic amplitudes in entrance and exit channel can be determined. The absolute spectroscopic factor of the 2~ neutron in the 3oSi ground state (entrance channel) is obtained by inserting the factor of the exit channel derived from the LCNO analysis of the 28Si-29Si scattering.
2. Experimental The experiments
were performed
at the Heidelberg
MP Tandem
at incident
beam
SCATTERING
OF Si
169
energies of 65 and 70 MeV, which is slightly above the Coulomb barrier. The negative ions for the beam were extracted from a Penning source with SiH, as discharge gas. Average beam intensities of 50 nA Si’+ were obtained behind the analyzing magnet. The targets consisted of SiO, on l*C backing, in which the isotopes 29Si and 3oSi, respectively, were enriched to more than 95 %. The thickness of silicon was about 80 pg/cm*. The main interest in this experiment was concentrated on the scattering at medium and backward angles where the interference due to the elastic transfer should be most pronounced. In elastic scattering of ions with similar masses, data points at these angles are obtained by detecting the recoil nuclei from the target at foreward angles. The cross section of the recoil nuclei measured in laboratory at an angle 6’ corresponds then to an elastic scattering cross section at (n-20) in the c.m. system. This procedure requires the separation of isotopes which can be achieved most conveniently by time-of-flight techniques in this mass region. The experiments were therefore performed with a time-of-flight spectrometer for heavy ions 19). The identification of the element number Z of the reaction products is not necessary, because all reactions which do not lead to silicon but to the masses of interest have highly negative Q-values. The time-zero detector of the spectrometer consists of a thin scintillator foil surrounded by a hemispherical mirror. The light emitted by a particle passing through the foil is detected by a fast photomultiplier. The foils were made from NE 111 with a thickness of 60 /lg/cm*. The second time signal is derived from the semiconductor detector at the end of the flight path which simultaneously measures the energy of the particle. Normally two energy detectors were mounted in a distance of 1” lab for which the same foil system acts as time-zero detector. Each of these detectors (400 mm* heavy ion series) covered a solid angle of 8.5 x 10m5 sr with a horizontal angular resolution of 0.5”. The length of the flight path was 155 cm. The time resolution of the spectrometer was below 300 ps. The limiting factor of this resolution is the high capacity of the semiconductor counters. Fig. I shows a display photo of the reaction products of the 28Si-29Si scattering. The energy of the particles is stored as a function of their time of flight in a twodimensional mode. The mass resolution is obtained by a cut through this matrix parallel to the time axis at given energy. Fig. 2 shows the mass resolution ofthe products of the 28Si-3 ‘Si scattering. The three silicon isotopes are clearly separated. The mass resolution is &4/A = 1.3 “/; (FWHM). Fig. 3 displays typical energy spectra of both the elastic scattered and recoil nuclei in the 28Si-29Si scattering. Recoil spectra have the advantage to be free from contamination lines such as the small but uncertain scattering on 28Si in the target. All points in the angular distributions at angles larger than 80” are taken from recoil spectra. The overall energy resolution in the experiments was of the order of 800 to 900 keV
170
K. D. HILDENBRAND
et al.
Fig. 1. Display photo of the reaction products of l*Si on 29Si at 70 MeV (8,, = 44.5”). The energy E of the particles is plotted against &--t, where t is the time of flight. The different masses are .4 = 32, 29, 28 (from left); A = 32 are recoil nuclei from the j2S target contamination.
depending on the telescope angle; 500 to 700 keV are due to kinematics and nearly 300 due to the target thickness. These values demonstrate clearly that in experiments such as the present one the intrinsic energy resolution (detector resoiution and energy straggling in the foil)ofthe system will play a minor role in most cases. The energy straggling n the foil is of the order of 20 % of the energy loss, which should be at least 1 MeV to
0 0
30
50
90
120
Et2
CHANNEF
Fig. 2. Mass spectrum of the reaction products of 2sSi on 3oSi at 70 MeV; 28Si and “Si are elastically scattered and recoil nuclei, “9Si comes from the reaction 3oSi(28Si, z’Si)29Si. The mass resolution for A = 28 is AA/A = 1.3 % FWHM.
SCATTERING
171
OF Si
‘“SE RECOIL @CM-107*
channei
Fig. 3. Energy spectra of ‘*Si and ‘%i taken at f&b = 36.5” in the elastic scattering from ?Z at E = 65 MeV. T he labels “2sSi” and “32S” in the upper spectrum denote the energy of particles scattered off the corresponding target contaminations (the isotopic enrichment of %3i was more than 95 %). The arrows 1 and 2 refer to the calculated position of the two first excited states of the outgoing particles.
700
b
J
k
100
r !-
10-I
10-2
~
10-3 ’ 00 ’ ’ 30”’ ’ ’ 60”’ ’ ’ 90”’ f ’ 120” Fig. 4. Angular
150”
180’
distributions of the 28Si-z9Si scattering at E = 65 and 70 MeV. The curves are LCNO calculations with the parameters listed in table 2.
172
K. D. NILDENBRAND
-
-....-.-.
et al.
.
1
‘“Si( zeS1,28Si)30Si CT,’u
F:
Fig. 5. Angular distributions of the 28Si-30Si scattering at E = 65 and 70 MeV together with LCNQ cafculations. The two neutrons were assumed to be transferred as a dineutron with spin zero between the scattered nuclei. The parameters used are fisted in table 2.
Fig. 6. Angular distributions of the neutron transfer reaction 30Si(28Sit t?5i)ZgSi at 70 MeV. The curve is a DWBA calculation, which takes into account the interference of two amplitudes due to the symmetric outgoing channel.
SCATTERING
OF
173
Si
19). The total energy resolution was sufficient to provide a reasonable time resolution separate the lowest excited state in 29Si at 1.28 MeV from the ground state. The relative normalization was given by the integrated beam current. The uncertainty is estimated to be smaller than 5 “/,, confirmed by a large number of points measured several times. Within this error the target thickness remained constant during the experiments. No evaporation of the SiO, layer was recognized. Absolute cross sections were obtained by normalizing the elastic scattering at small angles to Rutherford scattering. The error should be smaller than 10 “/,. Error bars in the drawings give only the statistical errors whenever they exceed the size of the points. The measured angular distributions of the 28Si-29Si and of the 2sSi-30Si scattering at 65 and 70 MeV are displayed in figs. 4 and 5, respectively. Fig. 6 shows the distribution of the neutron transfer 3oSi(28Si, 29Si)29Si at 70 MeV.
3. Results and reaction mechanism of the elastic neutron transfer 3.1. THEORETICAL
DESCRIPTION
The first practicable model worked out for the description of the coherent contribution of an elastic transfer was the LCNO model, which has been described in detail elsewhere 4Pr2, 13). Th ere f ore only the main ideas will be discussed here. The basic equation of the LCNO model can be written as follows r3): ‘CG + U(R) + (-)‘KX&)IXL(R)
= -QI_(R).
(3.1)
This equation is valid for the scattering of two heavy ions with identical boson cores (core spin = 0) and a valence particle with spinj 5 4. This is the case in the scattering of 28Si on 29Si and 3oSi where 2s, neutrons are exchanged between 28Sio+ cores. The kinetic energy of the’two cores is denoted by TR; U(R)is the optical potential and xL(R) is the wave function for the relative motion of the cores with orbital angular momentum L. The exchange potential Vexch (R), which is commonly called exchange integral J(R) [ref. ‘“)I, IS g’tven within the LCNO model by I/,,,,(R)
= J(R) = 1 &(y)kYy)4,(r
- R)d3r.
(3.2)
Here, 4r is the bound state wave function of the valence particle and V(r) the interaction between valence particle and core. The vector R denotes the distance between the cores, Y the distance between valence particle and core. The solution of eq. (3.1) gives the total scattering amplitudef(8). This amplitude may be interpreted as the superposition of the “normal” elastic scattering amplitude f,,(0) (calculated without exchange potential) and the transfer amplitude j;,(rr - f3). f(e)
= f,,(e) + ftr(n - 0).
Due to the change of sign of the exchange
potential
(3.3)
in eq. (3.1) the even partial waves
174
K. D. HILDENBRAND
et al.
are submitted to a more attractive (or more repulsive, depending on the sign of Ysxeh (R)) potential than the odd ones. Even or odd partial waves are therefore more suppressed, which leads to the characteristic L-staggering of the reflection coefficients is’). The reflection coefficients of the less absorbed partial waves are larger and will dominate in the total scattering amplitude. 3.2. CALCULATIONS
The LCNC? made1 requires the optical potential U(R) to describe the normal elastic scattering of the cores. The parameters of this potential can be changed together with the parameters of the exchange integral J(R) to get the best multiparameter fit to the measured data. In order to reduce these number of parameters, U(R) was independently determined by an optical model analysis of 28Si-28Si eta& scattering angular distrjbutjons at E fab = 50,60,67,74 and 77 MeV measured by Ferguson et al. ’ “). Fig. 7 shows the data together with the calculated curves (solid lines). At all energies the measured angular distributions are very well reproduced by the calculations using strongly absorbing potentials, as can be expected from the analysis 20) within the framework of the Biair model, The optical model parameters (WoodsSaxon potential) are listed in table 1. The sets used for the different energies are nearly identical. The angular distributions are symmetric around 90”, because the symmetrized amplitude, which can be considered as a superposition of a foreward and backward scattering amplitude (&(8) andf,,(z-8), respectively), contains only even I-values. The cross section of the foreward scattering, i.e. the square of the unsymmetrized amplitudef,l(8), shows a smooth behaviour (dotted curve in fig. 7). This cross section is shown as G/C~ in fig. 8. At all energies the curves exhibit the smoothly decreasing diffraction pattern of Fresnel type, becoming deeper with increasing energy. This Fresnel diffraction is typical of systems with Coulomb parameter y > I [ref. “)“J. It is evident that a normal optical model calculation for the scattering on 2”Si would predict similar smooth angular distributions. A reasonable change of the parameters cannot reproduce the strong oscillations seen in experiment. These oscillations and moreover some general features of the data indicate that a strong
&l,
cz,
t.21
0.49 0.49 0.49 0.49 0.49
1.20 I.21 1.20 1.21
(MYV)
2 2 7 14 17
A,
1.28 1.28 1.28 I.28 i 28
p____1_
&I
0.39 0.38 0.39 0.39 0.39
SCATTERING
ELASTIC
OF Si
SCATTERING
Data
by
175
‘*S, -‘*Si
AJFerguson
et al
E,,,= 50 MeV _.
Elkg=60 MeV
E,,,= 67 MeV
E,,.=7&
MeV ~
EL,,=77 MeV 1
4
”
LO”
11
’
60”
1
”
“1
80"
1
’
100"
1
I
’ 6Cl.4
Fig. 7. Optical mode1 calculations (solid curve) for zsSi -28Si Mott scattering angular distributions. The data are taken from ref. 20). The dotted curves show the unsymmetrized cross section of the corresponding calculations. The optical model parameters are listed in table 1.
176
K. D. HILDENBRAND
616,/
Elastic
et a/.
Scattering 28 SI-
28 SI
16’ I
30 Fig. 8. Unsymmetrized
angular
60
distributions
90
120
of the z8Si-z8Si
150 scattering
‘c M 20) (compare
fig. 7), plotted
coherent contribution of an elastic neutron transfer has to be taken into account. Thus LCNO calculations have been performed for all measured elastic scattering angular distributions. Once the optical mode1 parameters are fixed, the LCNO model leaves only the parameter SN, which can be considered as the strength of the exchange potential. The term SN contains the spectroscopic amplitude of the transferred particle. This will be discussed in detail in sect. 5. The LCNO calculations for the 28Si-29Si system are represented by solid lines in fig. 4. At both measured energies very good agreement was achieved with identical values for SN. The optical model parameters of both curves are identical, only the imaginary potential was raised for the higher energy, as has been done for the 28Si-28Si case. In the calculations for the 28Si-30Si system the two 2s, neutrons were assumed to be transferred as a dineutron. This allows for the same treatment within the LCNO model as in the case of a one-nucleon transfer. Again measured data and calculations are in agreement (fig. 5). The value of SN had to be assumed to be different for the two energies. The parameters of all LCNO calculations are listed in table 2. The assumption of an elastic neutron transfer leads to a quantitative understanding of all measured angular distributions. Additionally, this assumption is in accordance with some general features of the data: (i) In both systems the phase of the oscillations does not shift with the energy. This is known to be typical of an interference structure, where both amplitudes are slowly
SCATTERING
OF
177
Si
TABLE 2 Parameters
of LCNO
calculations
for
28Si-29Siand z8Si-30Si elastic scattering SN
(MT”) 28Si_29Si
65 70
100 100
1.20 1.20
0.49 0.49
5 10
1.28 1.28
0.49 0.49
28Si_30Si
65 70
100 100
1.20 1.20
0.52 0.46
8 10
1.28 1.28
0.39 0.39
4.0 4.0 220 198
varying with energy. This feature is therefore well described by the calculations without a change of the parameters. Other attempts using modifications of the optical model such as I-dependent imaginary potentials or “shallow potentials” fail in describing this feature. They fail in producing the deep oscillations, or significant changes of the parameters have to be made in order to get the phase fixed at different energies ‘l). (ii) The LCNO model describes very well the variation of the strength of the oscillations: In both systems oscillations become deeper with increasing energy at angles between 90” and 120”. On the left hand side of fig. 9 elastic scattering and transfer cross section of the 28Si-29Si scattering are shown separately at four different energies. Their coherent superposition, calculated with LCNO model [compare eq. (3.3)] is shown on the right hand side. The calculations were done with the best fit parameters of the measured angular distributions. At all energies the elastic scattering shows the expected smoothly decreasing behaviour. The maximum of the transfer amplitude shifts as a function of n-0 with increasing energy to larger angles, since the grazing angle eg., where the transfer is expected to have its maximum, shifts to more foreward angles. Thus, at the lowest energies there are structures oscillating around o/cR = 1.0 already below 90”. At 70 MeV both amplitudes have comparable heights between 90” and 120” which yields the strong oscillations seen in experiment. At 75 MeV the transfer amplitude has shifted so far, that the superposition shows a smooth decreasing below 90” followed by a separated transfer peak. 4. The neutron transfer 3oSi(28Si, 29Si)29Si In competition with the elastic two-neutron transfer the one-neutron transfer 3oSi(28Si, 29Si)29Si was observed in the scattering on 3oSi at 70 MeV, whereas it had not yet occurred at 65 MeV. Fig. 6 shows the angular distribution which is symmetric around 90” +. The black points are measured, the open ones were obtained by plotting measured points at n-0. ’ The angular distribution of this reaction has already been published in a contribution to the Argonne Symposium on heavy ion transfer reactions, held in March 1973 [see ref. ‘)I. Unfortunately, the shape of the angular distribution shown there is incorrect due to an error in a new evaluation program.
Fig. 9. LCNO calculations fur the 28Si~3gSi scattering at four different energies. On the left hand side the elastic scattering and the elastic transfer amplitude are plotted separately, the right hand side shows the coherent superposition of the two amplitudes. The change in the strength ot the interference oscillations is caused by a shift of the transfer amplitude to larger angles 0 with increasing energy.
SCATTERING
OF
Si
179
For calculating the cross section of this reaction, counting rates have been divided by a factor of 2 at each angle, because in every reaction two undistinguishable 29Si nuclei are created. The cross section is then defined as the probability per target nucleus that one projectile undergoes the reaction leading to that channel. With this definition the popular formula of the total cross section
is also valid for a symmetric outgoing channel. Of course this factor in the definition of the cross section has to be taken into account in a quantitative comparison of experimental and calculated cross sections. For evaluating the correct normalization factor the properly computed symmetric DWBA cross section has also to be divided by two. This has already been done in the factor given in subsect. 5.2. The solid line in fig. 6 represents a DWBA calculation, which takes into account the interference of forward and backward amplitude. This was achieved by modifying the multiple form factor option of the code DWUCK 22). The code calculates two form factors (which are identical in the present case of course) and two transfer amplitudes, one of them as a function of rc-0, and adds them coherently. For both entrance and exit channel the optical potential parametets of the 28Si-30Si scattering were used, bound-state parameters were y0 = 1.25 fm and a, = 0.65 fm (Woods-Saxon potential). The absolute height of the calculated curve was adjusted to the data, i.e. the relevant spectroscopic amplitudes were introduced (see sect. 5). The agreement between experiment and theory is fairly good. A change of the optical parameters did not improve the fit.
5. Spectroscopic 5.1. ELASTIC
TRANSFER
information
REACTIONS
The spectroscopic information on the transferred particles is contained in the exchange integral J(R) [see eq. (3.2)]. An analytical expression for J(R) is derived by using the approximation of Buttle and Goldfarb 23). Thus one obtains “)
J(R) = (SN)2 sexp(x3
cxR)/aR,
(5.1)
with a = 4 2M, Eb/h, where Ebis the binding energy of the transferred particle of mass MX. The quantity SNdetermines the height of J(R) and is used as the only free parameter which was varied to get the best fit for all measured data. It is the product of the spectroscopic amplitude S and a constant N which normalizes the relevant Hankel function to the bound-state wave function of the transferred particle. By calculating N for reasonable bound-state parameters it is possible to evaluate the spectroscopic
180
K. D. HILDENBRAND
factor S2. It should be mentioned, used for the calculation of N.
however,
et al.
that S2 then depends
on the parameters
The best fit for the 28Si-29Si scattering was obtained at both energies with SN = 4.0. With bound-state parameters y0 = 1.25 fm and a, = 0.65 fm one obtains a spectroscopic factor for the 2s, neutron in the 29Si ground state of S2 = 0.43. The experimental error of the determined value is small. A change of S within 10 % drastically varies the calculated cross section in the interference region between 90’ and 120” against that at lower energies. Hence, the error in S is determined by the error in the absolute cross section and the relative error between foreward and backward angles. Both errors are considered to be less than 10 ‘A (compare sect. 2). It has been discussed earlier r* ) that the measured value agrees very well with a reliable value from a ‘*Si(d, po)29Si experiment r4). Two spectroscopic factors from the reverse reaction 29Si(p, d,)‘*Si are reported. Both analyses at 27.5 MeV [ref. ““)I and 35 MeV [ref. ‘“)I gave S2 = 0.45 which also agrees well with the value given here. In a pure shell model picture which assumes a closed d, shell for the ‘*Si core one would expect S2 = 1.O for a neutron transfer leading to the ’ 9Si ground state. Older shell model calculations ’ 6 ) for sd shell nuclei actually predict this value starting from the assumption of an inert ‘*Si nucleus. It is known, however, that ‘*Si nuclei just as other nuclei in this mass region are deformed 26) which is considered to be the reason for the present difficulties of a unified interpretation of their properties r “). More recent calculations 1‘) yielded a spectroscopic factor S2 = 0.5 which is in agreement with the measured values. The elastic transfer in the ‘*Si -3 ‘Si scattering was treated as a transfer of a dineutron with spin zero. To achieve a good agreement with the data different values of SN had to be chosen in the calculations for the two measured energies. This effect was already observed in the scattering of 160 on ‘*O [ref. “)I. A possible explanation could be that the dineutron approximation fails at the higher energy and other processes may give a more important contribution. The deduced spectroscopic factors were S2 = 0.48 at 65 MeV and S2 = 0.39 at 70 MeV. 5.2. ONE-NEUTRON
TRANSFER
3oSi(28Si, 29Si)29Si
The spectroscopic amplitude is contained in the factor which adjusts the calculated curve to the data. In the formalism of the code DWUCK “) the calculated and the experimental cross section of the reaction A(a, b)B are related by do -=_ d&x, where JA and JB mentum (J, = 0, amplitudes of the The coefficient
2J,+l 2J,+l
x 10-4s~s:
B,,j do ~ ___ 2j + 1 dQn,e,
)
(5.2)
denote the spin of the nuclei, j the transferred total angular moJB = j = 4 in the present case); Si and St are the spectroscopic transferred neutron in entrance and exit channel, respectively. Blsj (1 = 0, s = j = f are the quantum numbers of the transferred
SCATTERING
181
OF Si
“) of the DWUCK form factor 22) B,,j neutron) can be obtained by a comparison frsj(r) and the factor used by Buttle and Goldfarb 23). This results in Blsj
firj(r)
m case with A,, z (h2/2Mx)(Nilcli) Here, Ni and Nr are the factors to wave functions in entrance and exit form factor behaves at large radii as
=
(4n)‘Nf
Ali
h,i(
iai
r),
(5.3)
of a neutron transfer with I = 0 [ref. ““)I. normalize a Hankel function to the bound-state channel. The radial partf,,j(r) of the DWUCK Ni h,,(ia,r). Therefore
Blsj = (47@N,h2/2M,ai (Ni = 6.1 fm-*,
= 668.2
xi = 0.7 fm-‘).
(5.4)
Inserting this value into eq. (5.2) as well as the factor da,,,/damvB, = 14.0 which gave the best overall agreement between theory and experiment one obtains S; S; = 0.3136. With St = 0.43 extracted from the analysis of the 28Si-29Si elastic transfer the spectroscopic factor Sf for the 2s+ neutron in the 30Si’ground state is S2 = 0.73. It should be pointed out again, that Sf Sf” depends on the product N’NF of the normalization constants and therefore sensitively on the geometry of the potential used for the calculation of the bound-state wave functions. transition to 29Si can be comThe deduced value S2 = 0.73 for the ground-state pared with the results obtained from neutron pickup reactions on 3oSi. (3He, E) measurements on 3oSi gave S2 = 0.7 [at 8 MeV, ref. ‘“) and 33 MeV, ref. 2 ‘) ] and S2 = 0.8 [at 7 MeV, ref. ““)I. The 30Si(d, t) reaction at 22.5 MeV also yielded S2 = 0.8, ref. 29), the (p, d) reaction at 27.3 MeV, S2 = 0.68, ref. ‘“). Somewhat lower is the result of the stripping reaction 29Si(d, p)30Si at 16 MeV with S2 = 0.45, ref. 30). As a conclusion one can say that the deduced spectroscopic factor just as the factor obtained from the elastic transfer on ’ 9Si agrees very well with factors deduced from light ion induced reactions.
6. Cross properties and conclusions Fig. 10 displays a comparison of angular distributions of all systems discussed: The elastic scattering of 28Si on 28Si [ref. ““)I, 29Si and 3oSi and the reaction 3oSi (28Si, 29Si)29Si. Two gross properties are obvious: (i) All distrib u t ions show a strong interference pattern. The width of the oscillations is 18O”/v] E 9” [ref. ‘)I. The Coulomb parameter yeis about 20 in all systems. Because it is a function of Ef, it varies only slowly in the energy region considered. (ii) The relative phase (maximum or minimum at 90°) is different in the elastic neutron transfer reactions and in the two systems with symmetric outgoing channel.
182
K. D. HILDENBRAND ELASTIC r?-‘-y
SCATTERING
/ ~, I,
OF
er al. SI
-ISOTOPES
ItT-,,
”
_!
Fig. 10. Comparison of angular distributions of the scattering of 28Si on 28~*9~30Si and of the is about 9”,.,,. in all systems; the relative reaction 3oSi(28Si, 29Si)z9Si. The width of the oscillations phase (maximum or minimum at 90”) depends on the symmetry of each system.
The width of the oscillations in the 28Si-28Si scattering results from the interference term in the classical Mott formula 31) which occurs because of the symmetrization of the total amplitude da/d!2
= IF(O)1’ = IL4(@+f&-W.
The same argument is valid for the reaction 3oSi(28Si, 29Si)29Si with the difference that the total amplitude has to be antisymmetrized, because the outgoing particles are fermions.
SCATTERING
The occurrence comes evident,
of the same width in the elastic transfer
when the quasiclassical
is used ‘), containing the transfer with the elastic scattering amplitude d4d.Q
183
OF Si
angular
distributions
be-
expression,
probability P,,(e). The coherent yields a cross section
superposition
= If,,(e) + P,,(n - e)f,,(n - W
This cross section will obviously show the same width of the interference structure as the Mott scattering, because the formula contains an analogue interference term. Because of the factor Ptr(e) the angular distribution is no longer symmetric. These considerations convincingly demonstrate that the oscillations in the scattering of 28Si on 29, 3oSi are due to the same effect as in the two other systems, i.e. the interference of two amplitudes. The relative phase of all four systems depends on their symmetry. The symmetrized amplitude of the 28Si-28Si Mott scattering system contains only even Z-values and therefore shows a maximum at 90”. The same is true of the reaction 3oSi(28Si, 29Si) 29Si. Assuming that the incoming channel spin (Si = 0) is conserved (negligible tensor force in the transfer potential) the two fermions in the outgoing channel have antiparallel spins (Sr = 0). Hence, only even partial waves occur, because the total wave function has to be antisymmetric. The phase of the system with particle exchange can be understood within the framework of the LCNO model. The zero spin of the cores and the exchange of s-nucleons determines that even scattering waves are submitted to an additional attractive potential (the exchange potential in eq. (3.1) appears with positive sign). Odd partial waves are therefore less absorbed and are dominating the scattering amplitude [see (3.1)]. Thus, the angular distributions of the scattering on 29Si and 3oSi are in antiphase to those of the two systems with symmetric outgoing channel.
References 1) H. Korer, Coulomb dominated transfer; in Argonne Physics Division Informal Report PHY1973 B (1973) D. Kovar, Higher energy transfer I, ibid. D. K. Scott, Higher energy transfer II, ibid. 2) N. Austern, Direct nuclear reaction theories (Wiley, New York, 1970) 3) W. E. Frahn, Ann. of Phys. 72 (1972) 524 4) H. G. Bohlen and W. von Oertzen, Phys. Lett. 37B (1972) 451 5) H. G. Bohlen, M. Feil, A. Gamp, B. Kohlmeyer, N. Maryuardt and W. von Oertzen, Phys. Lett. 41B (1972) 425 6) C. K. Gelbke, R. Bock, P. Braun-Munzinger, D. Fick, K. D. Hildenbrand, A. Richter and W. Weiss, Phys. Rev. Lett. 29 (1972) 1683 7) A. Gamp, W. von Oertzen, H. G. Bohlen, M. Feil, R. L. Walter and N. Marquardt, 2. Phys. 261 (1973) 283
184
K. D. HILDENBRAND
ef al.
8) C. K. Gelbke, R. Bock, P. Braun-Munzinger, D. Fick, K. D. Hildenbrand, W. Weiss, S. Wenneis and G. Baur, Phys. Lett. 43B (1973) 284 9) W. F. W. Schneider, Thesis, Universitat Marburg, 1972; B. Kohlmeyer, Thesis, Universitlt Marburg, 1973 10) W. F. W. Schneider, B. Kohlmeyer, F. Piihlhofer and R. Bock, Phys. Lett. 46B (1973) 195 11) G. Bauer and C. K. Gelbke, Nucl. Phys. A204 (1973) 138 12) W. von Oertzen, Nucl. Phys. Al48 (1970) 529 13) W. von Oertzen and W. Norenberg, Nucl. Phys. A207 (1973) 113 14) M. C. Mermaz, C. A. Whitten, J. W. Champlin, A. J. Howard and D. A. Bromley, Phys. Rev. C4 (1971) 1778 15) A. El-Naiem and R. Reif, Nucl. Phys. Al89 (1972) 305 16) P. W. Glaudemans, G. Wiechers and P. J. Brussaard, Nucl. Phys. 56 (1964) 548 17) B. H. Wildenthal and J. B. McGrory, Phys. Rev. C7 (1973) 714 18) K. D. Hildenbrand, R. Bock, H. G. Bohlen, P. Braun-Munzinger, D. Fick, C. K. Gelbke, W. von Oertzen and W. Weiss, Phys. Lett. 42B (1972) 425 19) C. K. Gelbke, K. D. Hildenbrand and R. Bock, Nucl. Instr. 95 (1971) 397 20) A. J. Ferguson, 0. Hausser, A. B. McDonald and T. K. Alexander, ANL report 7837 (1971) 187 21) C. K. Gelbke, R. Bock and A. Richter, Phys. Rev. C9 (1974) 852 22) P. D. Kunz, Algebra used by DWUCK, University of Colorado report CO?-535-613, unpublished, revised by S. Wenneis 23) P. J. A. Buttle and J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 24) F. Pellegrini, F. Gentilin, P. Guazzoni, S. Micheletti and M. Pinanelli, Phys. Rev. C2 (1970) 1440 25) G. J. Wagner, Nucl. Phys. Al76 (1971) 47 26) H. Rebel, G. W. Schweimer, G. Schatz, J. Specht. R. Lohken, G. Hauser, D. Habs and H. Klewe-Nebenius, Nucl. Phys. A182 (1972) 145 27) W. Bohne, C. K. Gelbke, P. Braun-Munzinger, W. Grochulski, H. L. Harney and H. Oeschler, Nucl. Phys. A222 (1974) 117 28) J. E. McQueen, Jr., J. M. Joyce and E. J. Ludwig, Nucl. Phys. Al47 (1970) 81 29) D. Dehnhard and Y. L. Yntema, Phys. Rev. 163 (1967) 1198 30) H. Mackh, H. Oeschler, G. J. Wagner, D. Dehnhard and H. Ohnuma, Nucl. Phys. A202 (1973) 497 31) A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1964) p. 608
I
2.N
Nuclear
Physics
A234 (1974)
Not to be reproduced
167-l
by photoprint
84; @ North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
ELASTIC NEUTRON TRANSFER IN THE SCATTERING OF Si ISOTOPES K. D. HILDENBRAND, R. BOCK t, H. G. BOHLEN, P. BRAUN-MUNZINGER, D. FICK, C. K. GELBKE, URSULA WEISS and W. WEISS Max-Planck-Institut
.ftir Kerphysik, Received (Revised
6 May
Heidelberg,
Germany
1974
4 July 1974)
Angular distributions of the elastic scattering of “Si on 29Si and 3oSi have been measured for incident beam energies at E = 65 and 70 MeV with a time-of-flight spectrometer for heavy 3oSi(28Si, 29Si)z9Si was observed in addition to the ions. At 70 MeV the neutron transfer elastic channel. The pronounced oscillations in the elastic scattering distributions are interpreted as being due to an elastic transfer of neutrons between the colliding nuclei during the scattering process. This assumption is in accordance with some general features of the data and allows for the extraction of spectroscopic factors of the transferred neutrons.
Abstract:
E
NUCLEAR REACTIONS *9*30Si(28Si, “Si), MeV; measured n(0). 29* 3oSi levels deduced
E = 65, 70 MeV; 3oSi(28Si, z9Si), E = 70 spectroscopic factors. Enriched targets.
1. Introduction In the last few years there has been a growing interest in studying few nucleon transfer reactions induced by heavy ions. This interest has been stimulated by the fact that it has proved to be possible to deduce spectroscopic information (via DWBA codes which are suited for heavy ion reactions) comparable to that obtained by light ion induced reactions ‘). The dominant physical features of heavy ion collision are Coulomb interaction and strong absorption of the projectiles ‘). As a consequence, the angular distributions of the elastic scattering exhibit the well known Fresnel diffraction pattern 3), which can be well understood in the framework of models taking into account the strong absorption either by a corresponding optical potential or more directly by a smooth cut-off parametrization of the reflection coefficients. Such a description, however, does not provide any nuclear structure information. This information can be obtained from the deviations from the normal diffraction scattering. One of the effects which yield such deviations is the elastic transfer. This special kind of transfer reproduces the particles of the entrance channel and can be written as follows: A+ (A+x) -+ (A+x)+ A. The amplitude of such a transfer will contribute coherently to that of the elastic scattering, causing structures in the angular distribution, the strength of which depends on the height of the transfer amplitude. + Present address: Gesellschaft fiir Schwerionenforschung, Darmstadt, W.-Germany. 167
168
It is evident
K. D. HILDENBRAND
that the study of such a transfer
et al.
combines
several advantages:
it can
be studied by measuring the elastic scattering with normally large cross sections instead of bell-shaped transfer angular distributions. Therefore nuclear structure information such as spectroscopic factors can be obtained directly from the elastic scattering. An additional advantage is the fact that the transferred particles are bound in identical states in entrance and exit channel. Therefore only one spectroscopic factor influences the cross section contrary to other heavy ion transfer reactions, where the information is contained in a product of two spectroscopic amplitudes. Additionally, in heavy ion reactions at energies in the vicinity of the Coulomb barrier only the outer parts of the wave functions have to be known. This situation is similar to sub-Coulomb stripping reactions, which, however, have small cross sections in most cases ‘). During the last two years the elastic scattering of nuclei which differ only by a few nucleons has been investigated mainly with projectiles between 12C and 19F In all cases the shape of the measured angular distributions could be [refs. ‘-r’)]. reproduced very well by the assumption of an elastic transfer. For a quantitative description of this transfer, which is necessary for deriving spectroscopic information, two models can be applied: The treatment within the framework of DWBA theory “) and the LCNO model 12, 13). It was proved that both models are equivalent as long as the interaction which causes the transfer can be treated in first order *I). Throughout the interpretation of the present data the LCNO model has been applied because it was the only model available at the beginning of the analysis. In the experiments reported here the elastic scattering of isotopes, i.e. nuclei differing only by the neutron number was studied for the first time in the mass region above A = 20. The silicon isotopes A = 28, 29, 30 were chosen because the additional neutrons of the heavier isotopes are 2s, neutrons. The formalism of the elastic transfer models is much easier if the transferred nucleons are bound to the cores with I = 0. For spectroscopic reasons this experiment seemed to be interesting because the spectroscopic factors of the 2~ neutron in the 29Si ground state determined with (d, p) reactions ’ 4, 1 j) disagree as well as the theoretical predictions [refs. ’ 6, “)I. The results of the 28Si-29Si scattering have partly been described in a previous publication I’). Besides the elastic scattering of 28Si on 3 ‘Si at 70 MeV the neutron transfer channel transfer 3oSi(28Si, * 9Si)2 9Si was observed. By a DWBA analysis of this “nonelastic” only the product of the spectroscopic amplitudes in entrance and exit channel can be determined. The absolute spectroscopic factor of the 2~ neutron in the 3oSi ground state (entrance channel) is obtained by inserting the factor of the exit channel derived from the LCNO analysis of the 28Si-29Si scattering.
2. Experimental The experiments
were performed
at the Heidelberg
MP Tandem
at incident
beam
SCATTERING
OF Si
169
energies of 65 and 70 MeV, which is slightly above the Coulomb barrier. The negative ions for the beam were extracted from a Penning source with SiH, as discharge gas. Average beam intensities of 50 nA Si’+ were obtained behind the analyzing magnet. The targets consisted of SiO, on l*C backing, in which the isotopes 29Si and 3oSi, respectively, were enriched to more than 95 %. The thickness of silicon was about 80 pg/cm*. The main interest in this experiment was concentrated on the scattering at medium and backward angles where the interference due to the elastic transfer should be most pronounced. In elastic scattering of ions with similar masses, data points at these angles are obtained by detecting the recoil nuclei from the target at foreward angles. The cross section of the recoil nuclei measured in laboratory at an angle 6’ corresponds then to an elastic scattering cross section at (n-20) in the c.m. system. This procedure requires the separation of isotopes which can be achieved most conveniently by time-of-flight techniques in this mass region. The experiments were therefore performed with a time-of-flight spectrometer for heavy ions 19). The identification of the element number Z of the reaction products is not necessary, because all reactions which do not lead to silicon but to the masses of interest have highly negative Q-values. The time-zero detector of the spectrometer consists of a thin scintillator foil surrounded by a hemispherical mirror. The light emitted by a particle passing through the foil is detected by a fast photomultiplier. The foils were made from NE 111 with a thickness of 60 /lg/cm*. The second time signal is derived from the semiconductor detector at the end of the flight path which simultaneously measures the energy of the particle. Normally two energy detectors were mounted in a distance of 1” lab for which the same foil system acts as time-zero detector. Each of these detectors (400 mm* heavy ion series) covered a solid angle of 8.5 x 10m5 sr with a horizontal angular resolution of 0.5”. The length of the flight path was 155 cm. The time resolution of the spectrometer was below 300 ps. The limiting factor of this resolution is the high capacity of the semiconductor counters. Fig. I shows a display photo of the reaction products of the 28Si-29Si scattering. The energy of the particles is stored as a function of their time of flight in a twodimensional mode. The mass resolution is obtained by a cut through this matrix parallel to the time axis at given energy. Fig. 2 shows the mass resolution ofthe products of the 28Si-3 ‘Si scattering. The three silicon isotopes are clearly separated. The mass resolution is &4/A = 1.3 “/; (FWHM). Fig. 3 displays typical energy spectra of both the elastic scattered and recoil nuclei in the 28Si-29Si scattering. Recoil spectra have the advantage to be free from contamination lines such as the small but uncertain scattering on 28Si in the target. All points in the angular distributions at angles larger than 80” are taken from recoil spectra. The overall energy resolution in the experiments was of the order of 800 to 900 keV
170
K. D. HILDENBRAND
et al.
Fig. 1. Display photo of the reaction products of l*Si on 29Si at 70 MeV (8,, = 44.5”). The energy E of the particles is plotted against &--t, where t is the time of flight. The different masses are .4 = 32, 29, 28 (from left); A = 32 are recoil nuclei from the j2S target contamination.
depending on the telescope angle; 500 to 700 keV are due to kinematics and nearly 300 due to the target thickness. These values demonstrate clearly that in experiments such as the present one the intrinsic energy resolution (detector resoiution and energy straggling in the foil)ofthe system will play a minor role in most cases. The energy straggling n the foil is of the order of 20 % of the energy loss, which should be at least 1 MeV to
0 0
30
50
90
120
Et2
CHANNEF
Fig. 2. Mass spectrum of the reaction products of 2sSi on 3oSi at 70 MeV; 28Si and “Si are elastically scattered and recoil nuclei, “9Si comes from the reaction 3oSi(28Si, z’Si)29Si. The mass resolution for A = 28 is AA/A = 1.3 % FWHM.
SCATTERING
171
OF Si
‘“SE RECOIL @CM-107*
channei
Fig. 3. Energy spectra of ‘*Si and ‘%i taken at f&b = 36.5” in the elastic scattering from ?Z at E = 65 MeV. T he labels “2sSi” and “32S” in the upper spectrum denote the energy of particles scattered off the corresponding target contaminations (the isotopic enrichment of %3i was more than 95 %). The arrows 1 and 2 refer to the calculated position of the two first excited states of the outgoing particles.
700
b
J
k
100
r !-
10-I
10-2
~
10-3 ’ 00 ’ ’ 30”’ ’ ’ 60”’ ’ ’ 90”’ f ’ 120” Fig. 4. Angular
150”
180’
distributions of the 28Si-z9Si scattering at E = 65 and 70 MeV. The curves are LCNO calculations with the parameters listed in table 2.
172
K. D. NILDENBRAND
-
-....-.-.
et al.
.
1
‘“Si( zeS1,28Si)30Si CT,’u
F:
Fig. 5. Angular distributions of the 28Si-30Si scattering at E = 65 and 70 MeV together with LCNQ cafculations. The two neutrons were assumed to be transferred as a dineutron with spin zero between the scattered nuclei. The parameters used are fisted in table 2.
Fig. 6. Angular distributions of the neutron transfer reaction 30Si(28Sit t?5i)ZgSi at 70 MeV. The curve is a DWBA calculation, which takes into account the interference of two amplitudes due to the symmetric outgoing channel.
SCATTERING
OF
173
Si
19). The total energy resolution was sufficient to provide a reasonable time resolution separate the lowest excited state in 29Si at 1.28 MeV from the ground state. The relative normalization was given by the integrated beam current. The uncertainty is estimated to be smaller than 5 “/,, confirmed by a large number of points measured several times. Within this error the target thickness remained constant during the experiments. No evaporation of the SiO, layer was recognized. Absolute cross sections were obtained by normalizing the elastic scattering at small angles to Rutherford scattering. The error should be smaller than 10 “/,. Error bars in the drawings give only the statistical errors whenever they exceed the size of the points. The measured angular distributions of the 28Si-29Si and of the 2sSi-30Si scattering at 65 and 70 MeV are displayed in figs. 4 and 5, respectively. Fig. 6 shows the distribution of the neutron transfer 3oSi(28Si, 29Si)29Si at 70 MeV.
3. Results and reaction mechanism of the elastic neutron transfer 3.1. THEORETICAL
DESCRIPTION
The first practicable model worked out for the description of the coherent contribution of an elastic transfer was the LCNO model, which has been described in detail elsewhere 4Pr2, 13). Th ere f ore only the main ideas will be discussed here. The basic equation of the LCNO model can be written as follows r3): ‘CG + U(R) + (-)‘KX&)IXL(R)
= -QI_(R).
(3.1)
This equation is valid for the scattering of two heavy ions with identical boson cores (core spin = 0) and a valence particle with spinj 5 4. This is the case in the scattering of 28Si on 29Si and 3oSi where 2s, neutrons are exchanged between 28Sio+ cores. The kinetic energy of the’two cores is denoted by TR; U(R)is the optical potential and xL(R) is the wave function for the relative motion of the cores with orbital angular momentum L. The exchange potential Vexch (R), which is commonly called exchange integral J(R) [ref. ‘“)I, IS g’tven within the LCNO model by I/,,,,(R)
= J(R) = 1 &(y)kYy)4,(r
- R)d3r.
(3.2)
Here, 4r is the bound state wave function of the valence particle and V(r) the interaction between valence particle and core. The vector R denotes the distance between the cores, Y the distance between valence particle and core. The solution of eq. (3.1) gives the total scattering amplitudef(8). This amplitude may be interpreted as the superposition of the “normal” elastic scattering amplitude f,,(0) (calculated without exchange potential) and the transfer amplitude j;,(rr - f3). f(e)
= f,,(e) + ftr(n - 0).
Due to the change of sign of the exchange
potential
(3.3)
in eq. (3.1) the even partial waves
174
K. D. HILDENBRAND
et al.
are submitted to a more attractive (or more repulsive, depending on the sign of Ysxeh (R)) potential than the odd ones. Even or odd partial waves are therefore more suppressed, which leads to the characteristic L-staggering of the reflection coefficients is’). The reflection coefficients of the less absorbed partial waves are larger and will dominate in the total scattering amplitude. 3.2. CALCULATIONS
The LCNC? made1 requires the optical potential U(R) to describe the normal elastic scattering of the cores. The parameters of this potential can be changed together with the parameters of the exchange integral J(R) to get the best multiparameter fit to the measured data. In order to reduce these number of parameters, U(R) was independently determined by an optical model analysis of 28Si-28Si eta& scattering angular distrjbutjons at E fab = 50,60,67,74 and 77 MeV measured by Ferguson et al. ’ “). Fig. 7 shows the data together with the calculated curves (solid lines). At all energies the measured angular distributions are very well reproduced by the calculations using strongly absorbing potentials, as can be expected from the analysis 20) within the framework of the Biair model, The optical model parameters (WoodsSaxon potential) are listed in table 1. The sets used for the different energies are nearly identical. The angular distributions are symmetric around 90”, because the symmetrized amplitude, which can be considered as a superposition of a foreward and backward scattering amplitude (&(8) andf,,(z-8), respectively), contains only even I-values. The cross section of the foreward scattering, i.e. the square of the unsymmetrized amplitudef,l(8), shows a smooth behaviour (dotted curve in fig. 7). This cross section is shown as G/C~ in fig. 8. At all energies the curves exhibit the smoothly decreasing diffraction pattern of Fresnel type, becoming deeper with increasing energy. This Fresnel diffraction is typical of systems with Coulomb parameter y > I [ref. “)“J. It is evident that a normal optical model calculation for the scattering on 2”Si would predict similar smooth angular distributions. A reasonable change of the parameters cannot reproduce the strong oscillations seen in experiment. These oscillations and moreover some general features of the data indicate that a strong
&l,
cz,
t.21
0.49 0.49 0.49 0.49 0.49
1.20 I.21 1.20 1.21
(MYV)
2 2 7 14 17
A,
1.28 1.28 1.28 I.28 i 28
p____1_
&I
0.39 0.38 0.39 0.39 0.39
SCATTERING
ELASTIC
OF Si
SCATTERING
Data
by
175
‘*S, -‘*Si
AJFerguson
et al
E,,,= 50 MeV _.
Elkg=60 MeV
E,,,= 67 MeV
E,,.=7&
MeV ~
EL,,=77 MeV 1
4
”
LO”
11
’
60”
1
”
“1
80"
1
’
100"
1
I
’ 6Cl.4
Fig. 7. Optical mode1 calculations (solid curve) for zsSi -28Si Mott scattering angular distributions. The data are taken from ref. 20). The dotted curves show the unsymmetrized cross section of the corresponding calculations. The optical model parameters are listed in table 1.
176
K. D. HILDENBRAND
616,/
Elastic
et a/.
Scattering 28 SI-
28 SI
16’ I
30 Fig. 8. Unsymmetrized
angular
60
distributions
90
120
of the z8Si-z8Si
150 scattering
‘c M 20) (compare
fig. 7), plotted
coherent contribution of an elastic neutron transfer has to be taken into account. Thus LCNO calculations have been performed for all measured elastic scattering angular distributions. Once the optical mode1 parameters are fixed, the LCNO model leaves only the parameter SN, which can be considered as the strength of the exchange potential. The term SN contains the spectroscopic amplitude of the transferred particle. This will be discussed in detail in sect. 5. The LCNO calculations for the 28Si-29Si system are represented by solid lines in fig. 4. At both measured energies very good agreement was achieved with identical values for SN. The optical model parameters of both curves are identical, only the imaginary potential was raised for the higher energy, as has been done for the 28Si-28Si case. In the calculations for the 28Si-30Si system the two 2s, neutrons were assumed to be transferred as a dineutron. This allows for the same treatment within the LCNO model as in the case of a one-nucleon transfer. Again measured data and calculations are in agreement (fig. 5). The value of SN had to be assumed to be different for the two energies. The parameters of all LCNO calculations are listed in table 2. The assumption of an elastic neutron transfer leads to a quantitative understanding of all measured angular distributions. Additionally, this assumption is in accordance with some general features of the data: (i) In both systems the phase of the oscillations does not shift with the energy. This is known to be typical of an interference structure, where both amplitudes are slowly
SCATTERING
OF
177
Si
TABLE 2 Parameters
of LCNO
calculations
for
28Si-29Siand z8Si-30Si elastic scattering SN
(MT”) 28Si_29Si
65 70
100 100
1.20 1.20
0.49 0.49
5 10
1.28 1.28
0.49 0.49
28Si_30Si
65 70
100 100
1.20 1.20
0.52 0.46
8 10
1.28 1.28
0.39 0.39
4.0 4.0 220 198
varying with energy. This feature is therefore well described by the calculations without a change of the parameters. Other attempts using modifications of the optical model such as I-dependent imaginary potentials or “shallow potentials” fail in describing this feature. They fail in producing the deep oscillations, or significant changes of the parameters have to be made in order to get the phase fixed at different energies ‘l). (ii) The LCNO model describes very well the variation of the strength of the oscillations: In both systems oscillations become deeper with increasing energy at angles between 90” and 120”. On the left hand side of fig. 9 elastic scattering and transfer cross section of the 28Si-29Si scattering are shown separately at four different energies. Their coherent superposition, calculated with LCNO model [compare eq. (3.3)] is shown on the right hand side. The calculations were done with the best fit parameters of the measured angular distributions. At all energies the elastic scattering shows the expected smoothly decreasing behaviour. The maximum of the transfer amplitude shifts as a function of n-0 with increasing energy to larger angles, since the grazing angle eg., where the transfer is expected to have its maximum, shifts to more foreward angles. Thus, at the lowest energies there are structures oscillating around o/cR = 1.0 already below 90”. At 70 MeV both amplitudes have comparable heights between 90” and 120” which yields the strong oscillations seen in experiment. At 75 MeV the transfer amplitude has shifted so far, that the superposition shows a smooth decreasing below 90” followed by a separated transfer peak. 4. The neutron transfer 3oSi(28Si, 29Si)29Si In competition with the elastic two-neutron transfer the one-neutron transfer 3oSi(28Si, 29Si)29Si was observed in the scattering on 3oSi at 70 MeV, whereas it had not yet occurred at 65 MeV. Fig. 6 shows the angular distribution which is symmetric around 90” +. The black points are measured, the open ones were obtained by plotting measured points at n-0. ’ The angular distribution of this reaction has already been published in a contribution to the Argonne Symposium on heavy ion transfer reactions, held in March 1973 [see ref. ‘)I. Unfortunately, the shape of the angular distribution shown there is incorrect due to an error in a new evaluation program.
Fig. 9. LCNO calculations fur the 28Si~3gSi scattering at four different energies. On the left hand side the elastic scattering and the elastic transfer amplitude are plotted separately, the right hand side shows the coherent superposition of the two amplitudes. The change in the strength ot the interference oscillations is caused by a shift of the transfer amplitude to larger angles 0 with increasing energy.
SCATTERING
OF
Si
179
For calculating the cross section of this reaction, counting rates have been divided by a factor of 2 at each angle, because in every reaction two undistinguishable 29Si nuclei are created. The cross section is then defined as the probability per target nucleus that one projectile undergoes the reaction leading to that channel. With this definition the popular formula of the total cross section
is also valid for a symmetric outgoing channel. Of course this factor in the definition of the cross section has to be taken into account in a quantitative comparison of experimental and calculated cross sections. For evaluating the correct normalization factor the properly computed symmetric DWBA cross section has also to be divided by two. This has already been done in the factor given in subsect. 5.2. The solid line in fig. 6 represents a DWBA calculation, which takes into account the interference of forward and backward amplitude. This was achieved by modifying the multiple form factor option of the code DWUCK 22). The code calculates two form factors (which are identical in the present case of course) and two transfer amplitudes, one of them as a function of rc-0, and adds them coherently. For both entrance and exit channel the optical potential parametets of the 28Si-30Si scattering were used, bound-state parameters were y0 = 1.25 fm and a, = 0.65 fm (Woods-Saxon potential). The absolute height of the calculated curve was adjusted to the data, i.e. the relevant spectroscopic amplitudes were introduced (see sect. 5). The agreement between experiment and theory is fairly good. A change of the optical parameters did not improve the fit.
5. Spectroscopic 5.1. ELASTIC
TRANSFER
information
REACTIONS
The spectroscopic information on the transferred particles is contained in the exchange integral J(R) [see eq. (3.2)]. An analytical expression for J(R) is derived by using the approximation of Buttle and Goldfarb 23). Thus one obtains “)
J(R) = (SN)2 sexp(x3
cxR)/aR,
(5.1)
with a = 4 2M, Eb/h, where Ebis the binding energy of the transferred particle of mass MX. The quantity SNdetermines the height of J(R) and is used as the only free parameter which was varied to get the best fit for all measured data. It is the product of the spectroscopic amplitude S and a constant N which normalizes the relevant Hankel function to the bound-state wave function of the transferred particle. By calculating N for reasonable bound-state parameters it is possible to evaluate the spectroscopic
180
K. D. HILDENBRAND
factor S2. It should be mentioned, used for the calculation of N.
however,
et al.
that S2 then depends
on the parameters
The best fit for the 28Si-29Si scattering was obtained at both energies with SN = 4.0. With bound-state parameters y0 = 1.25 fm and a, = 0.65 fm one obtains a spectroscopic factor for the 2s, neutron in the 29Si ground state of S2 = 0.43. The experimental error of the determined value is small. A change of S within 10 % drastically varies the calculated cross section in the interference region between 90’ and 120” against that at lower energies. Hence, the error in S is determined by the error in the absolute cross section and the relative error between foreward and backward angles. Both errors are considered to be less than 10 ‘A (compare sect. 2). It has been discussed earlier r* ) that the measured value agrees very well with a reliable value from a ‘*Si(d, po)29Si experiment r4). Two spectroscopic factors from the reverse reaction 29Si(p, d,)‘*Si are reported. Both analyses at 27.5 MeV [ref. ““)I and 35 MeV [ref. ‘“)I gave S2 = 0.45 which also agrees well with the value given here. In a pure shell model picture which assumes a closed d, shell for the ‘*Si core one would expect S2 = 1.O for a neutron transfer leading to the ’ 9Si ground state. Older shell model calculations ’ 6 ) for sd shell nuclei actually predict this value starting from the assumption of an inert ‘*Si nucleus. It is known, however, that ‘*Si nuclei just as other nuclei in this mass region are deformed 26) which is considered to be the reason for the present difficulties of a unified interpretation of their properties r “). More recent calculations 1‘) yielded a spectroscopic factor S2 = 0.5 which is in agreement with the measured values. The elastic transfer in the ‘*Si -3 ‘Si scattering was treated as a transfer of a dineutron with spin zero. To achieve a good agreement with the data different values of SN had to be chosen in the calculations for the two measured energies. This effect was already observed in the scattering of 160 on ‘*O [ref. “)I. A possible explanation could be that the dineutron approximation fails at the higher energy and other processes may give a more important contribution. The deduced spectroscopic factors were S2 = 0.48 at 65 MeV and S2 = 0.39 at 70 MeV. 5.2. ONE-NEUTRON
TRANSFER
3oSi(28Si, 29Si)29Si
The spectroscopic amplitude is contained in the factor which adjusts the calculated curve to the data. In the formalism of the code DWUCK “) the calculated and the experimental cross section of the reaction A(a, b)B are related by do -=_ d&x, where JA and JB mentum (J, = 0, amplitudes of the The coefficient
2J,+l 2J,+l
x 10-4s~s:
B,,j do ~ ___ 2j + 1 dQn,e,
)
(5.2)
denote the spin of the nuclei, j the transferred total angular moJB = j = 4 in the present case); Si and St are the spectroscopic transferred neutron in entrance and exit channel, respectively. Blsj (1 = 0, s = j = f are the quantum numbers of the transferred
SCATTERING
181
OF Si
“) of the DWUCK form factor 22) B,,j neutron) can be obtained by a comparison frsj(r) and the factor used by Buttle and Goldfarb 23). This results in Blsj
firj(r)
m case with A,, z (h2/2Mx)(Nilcli) Here, Ni and Nr are the factors to wave functions in entrance and exit form factor behaves at large radii as
=
(4n)‘Nf
Ali
h,i(
iai
r),
(5.3)
of a neutron transfer with I = 0 [ref. ““)I. normalize a Hankel function to the bound-state channel. The radial partf,,j(r) of the DWUCK Ni h,,(ia,r). Therefore
Blsj = (47@N,h2/2M,ai (Ni = 6.1 fm-*,
= 668.2
xi = 0.7 fm-‘).
(5.4)
Inserting this value into eq. (5.2) as well as the factor da,,,/damvB, = 14.0 which gave the best overall agreement between theory and experiment one obtains S; S; = 0.3136. With St = 0.43 extracted from the analysis of the 28Si-29Si elastic transfer the spectroscopic factor Sf for the 2s+ neutron in the 30Si’ground state is S2 = 0.73. It should be pointed out again, that Sf Sf” depends on the product N’NF of the normalization constants and therefore sensitively on the geometry of the potential used for the calculation of the bound-state wave functions. transition to 29Si can be comThe deduced value S2 = 0.73 for the ground-state pared with the results obtained from neutron pickup reactions on 3oSi. (3He, E) measurements on 3oSi gave S2 = 0.7 [at 8 MeV, ref. ‘“) and 33 MeV, ref. 2 ‘) ] and S2 = 0.8 [at 7 MeV, ref. ““)I. The 30Si(d, t) reaction at 22.5 MeV also yielded S2 = 0.8, ref. 29), the (p, d) reaction at 27.3 MeV, S2 = 0.68, ref. ‘“). Somewhat lower is the result of the stripping reaction 29Si(d, p)30Si at 16 MeV with S2 = 0.45, ref. 30). As a conclusion one can say that the deduced spectroscopic factor just as the factor obtained from the elastic transfer on ’ 9Si agrees very well with factors deduced from light ion induced reactions.
6. Cross properties and conclusions Fig. 10 displays a comparison of angular distributions of all systems discussed: The elastic scattering of 28Si on 28Si [ref. ““)I, 29Si and 3oSi and the reaction 3oSi (28Si, 29Si)29Si. Two gross properties are obvious: (i) All distrib u t ions show a strong interference pattern. The width of the oscillations is 18O”/v] E 9” [ref. ‘)I. The Coulomb parameter yeis about 20 in all systems. Because it is a function of Ef, it varies only slowly in the energy region considered. (ii) The relative phase (maximum or minimum at 90°) is different in the elastic neutron transfer reactions and in the two systems with symmetric outgoing channel.
182
K. D. HILDENBRAND ELASTIC r?-‘-y
SCATTERING
/ ~, I,
OF
er al. SI
-ISOTOPES
ItT-,,
”
_!
Fig. 10. Comparison of angular distributions of the scattering of 28Si on 28~*9~30Si and of the is about 9”,.,,. in all systems; the relative reaction 3oSi(28Si, 29Si)z9Si. The width of the oscillations phase (maximum or minimum at 90”) depends on the symmetry of each system.
The width of the oscillations in the 28Si-28Si scattering results from the interference term in the classical Mott formula 31) which occurs because of the symmetrization of the total amplitude da/d!2
= IF(O)1’ = IL4(@+f&-W.
The same argument is valid for the reaction 3oSi(28Si, 29Si)29Si with the difference that the total amplitude has to be antisymmetrized, because the outgoing particles are fermions.
SCATTERING
The occurrence comes evident,
of the same width in the elastic transfer
when the quasiclassical
is used ‘), containing the transfer with the elastic scattering amplitude d4d.Q
183
OF Si
angular
distributions
be-
expression,
probability P,,(e). The coherent yields a cross section
superposition
= If,,(e) + P,,(n - e)f,,(n - W
This cross section will obviously show the same width of the interference structure as the Mott scattering, because the formula contains an analogue interference term. Because of the factor Ptr(e) the angular distribution is no longer symmetric. These considerations convincingly demonstrate that the oscillations in the scattering of 28Si on 29, 3oSi are due to the same effect as in the two other systems, i.e. the interference of two amplitudes. The relative phase of all four systems depends on their symmetry. The symmetrized amplitude of the 28Si-28Si Mott scattering system contains only even Z-values and therefore shows a maximum at 90”. The same is true of the reaction 3oSi(28Si, 29Si) 29Si. Assuming that the incoming channel spin (Si = 0) is conserved (negligible tensor force in the transfer potential) the two fermions in the outgoing channel have antiparallel spins (Sr = 0). Hence, only even partial waves occur, because the total wave function has to be antisymmetric. The phase of the system with particle exchange can be understood within the framework of the LCNO model. The zero spin of the cores and the exchange of s-nucleons determines that even scattering waves are submitted to an additional attractive potential (the exchange potential in eq. (3.1) appears with positive sign). Odd partial waves are therefore less absorbed and are dominating the scattering amplitude [see (3.1)]. Thus, the angular distributions of the scattering on 29Si and 3oSi are in antiphase to those of the two systems with symmetric outgoing channel.
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K. D. HILDENBRAND
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