1.E.1 : 2.L
Nuclear Physics Al l5 (1968) 79--96; ~0 North-Hoiland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE 4STi(~, ~') REACT ION
AND SYSTEMATICS OF OCTUPOLE STATES IN THE Ti ISOTOPES
A. M. BERNSTEIN, E. P. LIPPINCOTT *, G. T. SAMPLE and C. B. THORN Department of Physics and Laboratory for Nuclear Science it,
M1T, Cambridge, Massachusetts, USA
Received 1 April 1968
Abstract: The 48Ti(ct, ~') reaction has been studied at 31 MeV with 100 keV resolution and high statistics. Spin and parity assignments have been made on the basis of the systematic shapes of the differential cross sections and by comparison with distorted wave Born approximation cal- culations. These assignments have been made up to an excitation energy of 6.4 MeV. The magni- tudes of the cross sections have been related to electromagnetic transition rates by use of the vibrational model. Based on the assumption of pure lf~ configurations, the lowest 4 + and the second 2 + states should be excited by two-step processes only. The differential cross sections for these states do not support this hypothesis. The most interesting feature of the results is the identification of seven 3- states in 48Ti ranging in excitation energy from 3.36 to 6.33 MeV which vary in strength from 1 to 3 Weisskopf units. Three 3- states of comparable strength have been previously located in 4eTi and five 3- states in bTi. The strongest of these states has 3.5 Weiss- kopf units in ~STi and 7 Weisskopf units in ~Ti. The weakness of the octupole states in this region compared to the 26 Weisskopf units found for the lowest 3- state in 4Ca is surprising and may be difficult to reconcile with our understanding of current microscopic descriptions of this mode.
E NUCLEAR REACTIONS 4STi(ct, ct'), E = 31 ; measured tr(E=,, 0). tSTi deduced levels, J, zr. Enriched target.
As part o f a study of the inelastic scatter ing of et-particles conducted at the MIT
Cyc lo t ron 1,2), levels o f 48Ti have been investigated. Previous ly 4aTi has been in-
vest igated with the (~, 0~') [refs. 3,4)], (p, p,) [ref. 5)], (d, p) [ref. 6)], (d, ~) [ref. 7)],
(p, d) [ref. 8)] and (t, p) [ref. 9)] react ions and by fl- and y-decay studies 10-12).
Compared to the previous (c~, C) exper iments 3,4) at 43 and 44 MeV, the present
exper iment has better energy resolut ion.
The low-ly ing posi t ive-par i ty levels can be compared with the shel l -model calcula-
t ions of McCu l len , Bayman and Zamick (MBZ) 13). Select ion rules based on this
mode l predict that the second 2 + and first 4 + levels should be excited by second-order transit ions 14).
Present address: Battelle Memorial Institute, Pacific N.W. Laboratory, Richland, Wash. tt This work is supported in part through funds provided by the Atomic Energy Commission under
80 A. M. BERNSTEIN et al.
700 6 z
f . - . ~
48Ti(~z, off) REACTION 81
Another reason for interest in 48Ti was to continue the study of the systematics of octupole states. In particular the octupole strength has been found to be fractionated in the Ca isotopes 1) with three states found in 4Ca and 48Ca and six in 42Ca and 44Ca. It is interesting to examine the behavior of the octupole strength in 4STi where there are both neutrons and protons outside of closed shells.
2. Experimental procedure
The experimental apparatus has been described in a previous paper 1). Alpha par- ticles of 31 MeV energy were scattered by a thin foil (about 1 mg/cm 2) of metallic 48Ti enriched to 99 %. Scattered alpha particles were detected with 500/~m silicon surface-barrier detectors, and the resulting energy spectra were recorded on a 1024- channel analyser. The beam intensity was typically 0.2/~A, and over-all resolution was approximately 100 keV.
Spectra were taken at 1.8 intervals from 15 to 60 . A typical spectrum at an angle that favors negative-parity states is shown in fig. 1. The angular aperture was about 3o in the scattering plane. The relative angular accuracy is 0.2 , whereas the absolute angular accuracy is +_0.4 . The data were analysed by computer using a least-squares fitting procedure to give the cross section at each angle. Absolute cross sections were obtained by comparison to the previously measured 4Ca cross section at the 30 maximum with an overall error of 15 o/ /O"
The Q-values of the excited states were determined by assuming the first, third and sixth excited states to lie at 0.984, 2.423 and 3.365 MeV, respectively. This calibra- tion leads to an error of about 10 keV for each MeV of excitation.
3. Elastic scattering
The elastic scattering angular distribution is shown in fig. 2. The solid line is an optical-model fit to the data. A four-parameter complex potential well of the form
U (r) = - ( V + iW) [1 + e ('-R)/" ]-
was used and the parameters varied for the best least-squares fit. The parameters for the curve shown in fig. 2 are V = 56.3 MeV, W -- 13.0 MeV, R = 5.58 fm and a = 0.646 fm. These parameters were used to calculate the inelastic scattering cross sections. Other parameter sets that give equivalent fits were also obtained.
4. Inelastic scattering
4.1. SPIN AND PARITY ASSIGNMENTS
The inelastic scattering data obtained in this experiment were analysed using the DWBA in which the ingoing and outgoung s-particle waves were distorted by the
82 A, M, BERNSTEIN et al.
optical potential found from the elastic scattering t 5). The angular distributions are calculated by using the collective model in which the surface of the nucleus is vibrating
4~x 8Ti Elastic i03~ xx
, o -
i0 ~ -- i ~
Io-' I I I I I I0 20 30 40 50 60 70
Fig. 2. Elastic scattering cross section for 31 MeV 0c-particles f rom 48Ti. The solid l ine is an optical- model fit.
around a spherical equilibrium shape according to the formula (to first order)
R(0', ,p') = Ro[1 + Y~ %.r?(o', ~o')1. lm
The effect of this oscillation on the optical potential is to introduce non-spherical
4STi(~t, ~t') REACTION 83
terms which give rise to inelastic scattering. This theory has the advantage that the shape of the angular distribution is then determined without any additional param- eters. Although this theory leans on the vibrational model, it can be shown that the shapes of the predicted cross sections are sensitive only to the angular momentum transfer and not to the specific nuclear model employed.
As has been shown in detail in a previous (e, e') experiment in the Ca isotopes 1), the DWBA theory is in excellent agreement with the shapes of the observed differen- tial cross sections. These angular distributions have characteristic shapes for different angular momentum transfers so that spin and parity assignments can be made with confidence. The reliability of this procedure is indicated by the high degree of similari- ty of the shapes of the differential cross sections for states of the same final spin and parity.
The only free parameter in the DWBA theory is determined by the magnitude of the cross section. This parameter is (/~/)2, the root-mean square deformation of the ground state due to zero point oscillations. Assuming that neutrons and protons move in phase, as one expects for AT = 0 excitations, the value of /~ obtained from in- elastic scattering data can then be used to calculate an electromagnetic transition rate. It is assumed that the amplitude of vibration of the charge distribution and optical potentials are equal so that (/3R)E ~ = (/~R)~, where EM indicates the appropriate electromagnetic value and e the quantity measured in e-particle scattering 16).
In the past it has been customary to calculate the inferred electromagnetic transi- tion rate from the vibrations of a uniform charge distribution whose radius REM is 1.2 A ~ fm [ref. 1)]. This procedure is convenient because it leads to a simple formula. However the measured charge distributions ~7) in nuclei are not uniform but can be more accurately characterized by the Fermi shape
p(r) = p0[1 +e('-c)/~] -1, where on the average c = 1.08 A 6 fm and a = 0.585 fm [ref. 17)]. (This gives an equiv- alent 10 ~o to 90 % skin thickness of 2.5 fro.) The multipole transition rates calculated from a uniform-charge distribution compared to that calculated from a Fermi-charge distribution will be underestimated, particularly for the high multipolarities, where the surface region is highly weighted ~s). In the case of 48Ti, using the parameters just indicated, the ratios of the inferred electromagnetic transitions rates calculated by the use of a Fermi-charge distribution to that of a uniform-charge distribution are 1.14, 1.43 and 2.0 for multipolarities l of 2, 3 and 4, respectively. Therefore, the in- ferred electromagnetic transition rates have been calculated using the vibrations of a Fermi charge distribution. These rates which are presented in tables 1 and 2 are given in terms of a single-particle (Weisskopf) unit 19)
Bs.p.(El),O~l)--2l+l(3 ) 2 47r ~ (REM)2l e2 fm2/"
84 A. M. BERNSTEIN el al.
TABLE 1 Results for 2 + and 4 ~ states in aSTi
j= E(MeV) /?,~. G~ ~) Error in G~ (~)
2 + 0.984 0.2l 16 15 2.42 0.058 1.2 15 4.96 0.045 0.72 25
4 + 3.24 0.082 4.6 15 5.16 b) 0.036 0.9 25 5.34 ~') 0.051 1.8 30
a) The inferred electromagnetic transit ion rates in single-part