Comments on unhappiness factors in simultaneous transfer DWBA description of (t,p) and (p,t) reactions

Finite-range one-step (simultaneous) transfer DWBA calculations have been carried out for (t, p) and (p, t) reactions involving some light-and medium-heavy nuclei with closed shells + 2 valence neutrons. A Tang-Herndon triton was used as well as realistic nuclear overlap functions generated by a sturmian procedure from a Schrödinger equation for two interacting particles in an external potential. The effect of the latter is to enhance cross section magnitudes by factors of 2–3, some of this enhancement being due to use of a density-dependent effective two-body interaction. Compared with data considerable unhappiness (underprediction) factors, however, remain. This finding is at variance with results reported recently by other authors.     

Finite-range DWBA analysis of (p, α) reactions on Be and B

The ground state angular distributions of the reactions ⁹Be(p, α)⁶Li and ¹¹B(p, α)⁸Be are analyzed by means of the DWBA theory. One-step direct mechanisms only are considered, but for both reactions the inclusion of the heavy-particle pick-up is found to be essential. DWBA fits to the data are obtained (a) in the zero-range approximation, (b) in the “fixed-range” approximation and (c) by an exact finite-range calculation. Only the latter method yields both an acceptable fit and reasonable values for the spectroscopic factors.     

Energy levels in even actinide isotopes from (t,p) reactions

Energy levels in^{232, 234}Th,^{236, 238, 240}U and in²⁵⁰Cm have been measured using the (t, p) reaction. Angular distributions were obtained for the^{234, 238}U targets and evidence for second order effects in the direct reaction mechanism was found.     

A microscopic DWBA analysis of (6Li, 6He) reactions

The spin-isospin flip (⁶Li, ⁶He) reaction has been studied. The experiment was performed with a 34 MeV ⁶Li beam incident upon ^{25, 26}Mg and ²⁷Al targets. Angular distributions associated with the lowest $\text{5}\text{2}{}^{\mathrm{+}}\text{,}\text{1}\text{2}{}^{\mathrm{+}}\text{,}\text{7}\text{2}{}^{\mathrm{+}}{}^{25}\text{Al}\text{, 5}{}^{\mathrm{+}}\text{, 3}{}^{\mathrm{+}}\text{, 1}{}^{\mathrm{+}}{}^{26}\text{Al and}\text{5}\text{2}{}^{\mathrm{+}}{}^{27}\text{Si}$ states were measured. Microscopic DWBA calculations were performed using Yukawa central and r². Yukawa (r² · Y) tensor (μ = 1.0 fm^?1) potentials averaged over the ⁶Li(⁶He) cluster model wave functions. Knock-on (KO) exchange corrections corresponding to the central interaction are included using the local-energy approximation (LEA) method. The conclusion is that the quasi-elastic process is of major importance in the (⁶Li, ⁶He) transition studied in this work.     

Nucleon-nucleon final-state interactions in the reactions He(p, d)2p, He(p, t)2p and He(p, He)pn

The energy spectra of deuterons, tritons and ³He particles from the reactions ³He(p, d)2p, ⁴He(p, t)2p and ⁴He(p, ³He)pn have been measured at angles between 6° and 60° lab. The ³He(p, d)2p reaction was studied at both 30.5 and 49.5 MeV incident proton energies, while the other two reactions were studied at 49.5 MeV only. The energy spectra are compared with calculations based on the Watson-Migdal model of final-state interactions.     

Unnatural parity transitions in (p,t) reactions using realistic triton wavefunctions

The reaction ²⁰⁸Pb(p,t)²⁰⁶Pb leading to the unnatural parity 3⁺ state of ²⁰⁶Pb is analyzed in the framework of finite range DWBA formalism. The main aspect of the present analysis is the use of a realistic triton wavefunction with S, S′ and D state components which permit this transition in DWBA. Excellent agreement between theoretical prediction and experiment is obtained casting serious doubt on the treatment of this reaction as a sequential transfer process.     

The (p,t) reactions on nuclei in the rare earth region

The (p, t) reactions on isotopic targets of ^{178, 180}Hf and all the stable isotopes of Yb and on natural targets of Gd, Dy, Er, Hf, Ta, W, Os and Au were studied at a beam energy of 19 MeV with an average resolution of 12 keV. A split-pole magnetic spectrometer was used to measure (p, t) Q-values and absolute differential cross sections. On the basis of angular distribution shapes definite 0⁺ and tentative 2⁺ assignments were made. Rotational bands were identified assuming an I(I+1) spacing. The (p, t) reaction populates excited 0⁺ states strongly in ¹⁷⁴Yb, ¹⁷⁶Hf, ¹⁶⁶Yb and several Gd, Dy and Er isotopes. The ¹⁷⁴Yb and ¹⁷⁶Hf 0⁺ states are discussed in terms of the pairing phase transition and in terms of Nilsson orbitals with unequal (p, t) reaction amplitudes. Members of gamma and octupole vibrational bands were observed in the even-N nuclei. The lowest L = 0 transfers to states in ^{169, 171}Yb were found to have less than 55% of the strength to ground states in adjacent even-N nuclei. A strong L = 0 transfer to a state at 1513 keV in ¹⁷¹Yb indicates the presence of a possible K = 0 core vibration coupled to the unpaired $\text{5}\text{2}$[512] neutron. The natural targets have furnished information on trends in cross sections for members of ground bands, gamma bandheads, 3^? octupole states, and strongly excited 0⁺ states.     

The (d,t) and (d,t) reactions on 18O and the 1p spin-orbit splitting

The reactions ¹⁸O(d, t)¹⁷O and ¹⁸O(d, τ)¹⁷N have been investigated at ifE_{d = 52 MeV}. Energy spectra of tritons and τ particles have been measured up to excitation energies of 25 MeV in ¹⁷O and 12 MeV ¹⁷N, respectively, and spectroscopic factors have been obtained by a DWBA analysis of the measured angular distributions. From a comparison of the t-and τ-spectra the distribution of $\text{T =}\text{1}\text{2}\text{and}\text{3}\text{2}$ spectroscopic strengths in ¹⁷O could be deduced and analog relations between $\text{T =}\text{3}\text{2}$ states in ¹⁷N and ¹⁷O could be established. Nearly the total $\text{T =}\text{3}\text{2}$ strengths of the $\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{and}\text{1}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ shells and nearly the complete $\text{T =}\text{1}\text{2}$ strength of the $\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ shell have been found, whereas only one third of the $\text{T =}\text{1}\text{2}$ strength of the $\text{1}\text{p}\text{3}\text{2}$. Shell could be clearly identified. The observed centroid energies are understood from the different $\text{1}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$$\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?}}\text{1)}$ and $\text{1}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$ effective residual interactions. This supports a strong isospin dependence of the 1p spin-orbit splitting.     

Observation of two-hole states at high excitation energy in (p,t) reactions

Bumps of two-hole states at high excitation energies were observed systematically in the triton spectra from (p, t) reactions with 52 MeV protons on nuclei in a broad range of masses. The cross sections of the bumps are almost equal for various targets with the same deep major shells. These cross sections vary discontinuously with variation of the corresponding deep major shell. About 20 to 50 % of the total expected strength is observed experimentally, if the bumps are assumed to arise from two-neutron pickup from the deep major shells. The centres of gravity of the bumps are located at excitation energies of about 7 to 9 MeV in all cases. On the other hand, the widths of the bumps change from about 3 MeV for ⁶⁶Zn to about 9 MeV for ²³⁰Th.     

Q values in β+-decay, (p,n) and (3He,t) reactions

Atomic excitation corrections to Q-values in β⁺-decay determined by (p, n) and (³He, t) reactions are shown to be comparable to current experimental precision, but too small to account for observed discrepancies.     

The (p, t) and (p, τ) reactions on O at 39.8 MeV

Differential cross sections have been measured at forward angles for (p, t) and (p, τ) transitions from ¹⁷O to the ⁻ ground states and lowest-energy ⁻ states in the ¹⁵O and ¹⁵N mirror nuclei. The data are compared with DWBA calculations using simple single-particle and single-hole wave functions. When the (p, t) and (p, τ) transitions are considered separately, the calculated and experimental ratios of the ⁻ integrated cross sections to the ⁻ integrated cross sections agree to within 30 %; however, the ratios of (p, τ) cross sections to the mirror state (p, t) cross sections are calculated to be about twice as large as actually measured. This experimentally observed reduction of the (p, τ) cross section relative to the (p, t) cross section can possibly be attributed to interference between the S = 0 and S = 1 components of the (p, τ) transitions.     

Role of the RMS radius in DWBA calculations of the (p,d) reaction

The single-neutron pick-up reaction cross sections appear to be better correlated with the rms radius of the neutron orbit rather than with r₀, the radius parameter of the single-particle Woods-Saxon well. Form factors which correspond to particles bound by the experimental separation energies and which have rms radii for the $\text{2}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, 1}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{and}\text{1}\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ orbits obtained from Coulomb energy shifts, lead to spectroscopic factors and S_>/S_< ratio for the ^46,48,50Ti(p, d) reactions which are in a much better agreement with theoretical expectations than those obtained with constant r₀.     

12C(p,n)12N and 16O(p,n)16F reactions at Ep = 35 and 40 MeV: Reliability of the information obtained from DWBA analysis of low-energy (p,n) data

A high-resolution study of the ¹²C(p, n)¹²N and ¹⁶O(p, n)¹⁶F reactions was made at E_p = 35 and 40 MeV. The low-lying states in ¹²N(1⁺, 2⁺ and 2⁻) and in ¹⁶F(0⁻, 1⁻, 2⁻ and 3⁻) were clearly resolved, and their angular distributions were measured. Extensive DWBA analysis was made and compared with the data. The calculated angular distribution shapes are found to be in agreement with the data and insensitive to the choice of the parameters involved. On the other hand, the magnitudes of the DWBA cross sections depend strongly on the bound state parameters in the case of a transition from a tightly bound state to a loosley bound state. In the other cases the overall uncertainty of the DWBA cross section magnitudes was estimated to be about ±30%. Within this uncertainty the experimental cross sections for the ¹²C(p, n) reaction were explained by the calculation, but those for the ¹⁶O(p, n) reaction were not: the observed strengths were about a half of the calculated values. Since these results agree with those at intermediate energies, the origin of the discrepancy is considered to be in the structure of the mass 16 nuclei rather than in reaction dynamics. In general, the present results compare well with those at intermediate energies, indicating that the structure information extracted from low-energy, high-resolution (p, n) data is basically sound if careful analysis of the data is made.     

(p,t) strengths in the Ga isotopes

The (p, t) reaction on the two stable Ga isotopes has been performed at 25 MeV, with 11 keV energy resolution. Levels up to 3.5 MeV in ⁶⁷Ga and 4.3 MeV in ⁶⁹Ga were measured. Differences observed between the distribution of the L = 2 (p, t) strength and the distribution of the B(E2)↑ strength may be explained by a mixed character of the $\text{7}\text{2}{}_2$^? level wave function. The distribution of the L = 0 strength indicates that the striking change in the ground-state structure shown between N = 40 and 42 already begins, although weakly, between N = 38 and 40.     

Anomalous behavior of l = 0 transitions in Mo(d,p) and Mo(d,t) reactions

The states of ⁹⁹Mo dominantly populated by l = 0 transitions in ⁹⁸Mo(d,p) and in ¹⁰⁰Mo(d,t) are not the same. The ratio S(d,t)/S(d,p) for these states differs by a factor of 21 which is an order of magnitude larger than in any other known case. This represents a breakdown of the quasi-particle model.     

The (p-d-t) process in strong (p,t) transitions

The (p-d-t) process in ⁴⁸Ca(p, t)⁴⁸Ca(p, t)⁴⁶Ca(0⁺, g.s.; 2⁺, 1.34 MeV) and ¹¹⁶Sn(p, t)¹¹⁴Sn(0⁺, g.s.) at 20 MeV was investigated. The calculations showed that the two-step process is important in all the transitions investigated, and is enhanced in the Sn transition by the pairing correlations in the BCS ground states.     

Levels excited in73Ge by (d,p) and (p,p′) reactions

Angular distributions from the reactions⁷²Ge(d, p)⁷³Ge and⁷³Ge(p, p′)⁷³Ge have been studied with 8.0 MeV deuterons and 12.0 MeV protons, respectively, using the Aldermaston Tandem Van de Graaff generator and multi-angle magnetic spectrograph. The ground stateQ-value of the (d, p) reaction was measured to be 4.571±0.004 MeV. The energies, values of the transferred orbital angular momenta, parities and transition strengths in both reactions were determined by use of distorted-wave Bornapproximation calculations, for levels up to 3 MeV excitation energy. The sums of the transition strengths are compared with shell model calculations as well as with values obtained in neighbouring nuclei. Discrepancies between parity and possible spin assignments of states observed in the two reactions are discussed.     

A comparison of the He(p, 2p)d and He(p, pd)p reactions

The ³He(p, 2p)d and ³He(p, pd)p reactions have been compared at three bombarding energies from 65 to 100 MeV. A comparison of plane wave impulse approximation calculations to the experimental data indicates that multiple scattering effects are large and energy dependent but that they primarily produce a uniform reduction in cross section. Although multiple scattering effects are large the ratio of the cross sections for the two reactions is in agreement with that predicted by the impulse approximation.     

Excitation function for high-energy (p, 2p) reactions

The excitation functions of the (p, 2p) reaction cross sections for the elements ²⁵Mg, ³⁰Si, ⁵⁷Fe, ⁶⁸Zn, ¹¹⁸Sn, ¹⁴²Ce and ¹⁸⁶W have been calculated in the energy region above 100 MeV. The calculated values are compared with the available experimental results and the agreement between the two is found to be fairly good. The dependence of σ(p,2p) on the target mass number has been explained in terms of the expression derived.