Modified Diffraction Model Research Articles

Isospin triplet A=14: search for states with enhanced radii

This article is devoted to study of isobar-analogue states 1− in triplet A=14: 14C-14N-14O. Previously signs of neutron halo in the 1−, 6.09 MeV state of 14C were obtained by two independent groups. In this article we propose to study neighbouring nuclei 14N and 14O using the Modified diffraction model (MDM) method and the method of Asymptotic normalization coefficients (ANC). Methods were applied to experimental differential cross sections of 14C(α,α)14C scattering and reactions 13C(3He,d)14N and 14N(3He,t)14O. MDM and ANC gave practically similar within errors radii for the studied 1− states: the 6.09 MeV state in 14C – 2.7±0.1 fm, the 8.06 MeV state in 14N – 2.7 ± 0.1 fm, the 5.17 MeV state in 14O – 2.6 ± 0.2 fm. Moreover, the signs of proton halo in the 1− state of 14N were obtained for the first time.

Isospin triplet A=12: search for states with enhanced radii

Two independent methods: Asymptotic Normalization Coefficients (ANC) and Modified Diffraction Model (MDM) were applied to new and existing experimental data to search for states with enhanced radii in isobaric analogue states (IAS) of 12B and 12N. The ANC analysis of the 11B(d,p)12B reaction at Ed = 21.5 MeV has confirmed neutron halo existence for two states of 12B: 2−, 1.67 MeV and 1−, 2.62 MeV [1]. Some new results were obtained for higher excited states: halo-like states were observed for 0+, 2.72 MeV and 3−, 3.39 MeV states. It should be mentioned that the last one is unbound state, which is 19 keV above the neutron emission threshold and in both states last neutron has a non-zero orbital momentum (ln=1 and ln=2 correspondingly). We propose to use MDM to study isobaric analogue states of 12B in the mirror 12N nucleus and apply it to analyze the (3He,t) reaction data. The existing in the literature data are not completed enough to make definite conclusion about halo in the 2− and 1− states of 12N. This fact simulated us to carry out a new experiment on the 12C(3He,t)12N reaction at the end of 2018. The measurements were conducted at the University of Jyväskylä (Finland) using the K130 cyclotron to produce a 3He beam at E(3He)=40 MeV. The differential cross sections of the 12C(3He,t)12N reaction were measured in the c.m. angular range 8°–70°. Preliminary results for angular distributions are obtained. MDM analysis of preliminary (3He,t) experimental data is done.

States of the 12N Nucleus with Increased Radii

The differential cross sections of the $^{12}$C($^3$He,t)$^{12}$N reaction leading to formation of the 1$^+$ (ground state), 2$^+$(0.96 MeV), 2$^{-}$(1.19 MeV), and 1$^{-}$(1.80 MeV) states of $^{12}$N are measured at $E$($^3$He)=40 MeV. The analysis of the data is carried out within the modified diffraction model (MDM) and distorted wave Born approximation (DWBA). Enhanced $rms$ radii were obtained for the ground, 2$^{-}$(1.19 MeV), and 1$^{-}$(1.80 MeV) states. We revealed that $^{12}$B, $^{12}$N, and $^{12}$C in the IAS with T=1, and spin-parities 2$^{-}$ and 1$^{-}$ have increased radii and exhibit properties of neutron and proton halo states.

Exotic states of 13C nuclei

The new study of differential cross sections of 13C( α,α’)13C inelastic scattering at Eα=29 MeV with the excitation of the 9.90 MeV state of the initial nucleus has been carried out. Modified diffraction model analysis showed that the root-mean-square (RMS) radius of the nucleus at this level might be approximately 20% less than the one of the ground state. We have showed the possibility of the existence of various states in the structure of the 13C nucleus.

Investigation of exotic states of 13C at low energy

The differential cross-sections of the elastic and inelastic [Formula: see text]C scattering have been measured at E[Formula: see text][Formula: see text]MeV. The radii of the exited states: 3.09 [Formula: see text] and 8.86 [Formula: see text] MeV were determined using the Modified Diffraction Model. The radii of these excited states are larger than that of the ground state of [Formula: see text]C, confirming the suggestion that the 8.86 [Formula: see text] MeV state could be an analog of the Hoyle state in [Formula: see text]C and the 3.09 [Formula: see text] MeV state has a neutron halo. The possibility of coexistence of various exotic states in the structure of the [Formula: see text]C nucleus is shown.

The cluster states in light nuclei

The differential cross-sections of the elastic and inelastic α +11B scattering was measured at E(α) = 29 MeV. The analysis of the data by Modified Diffraction Model (MDM) showed that the root-mean- square (RMS) radius of the 11B state 3/2-, E* = 8.56 MeV is ~ 0.5 fm larger than that of the ground state. It is found that the radius of 3/2- (8.56 MeV) state in the 11B nucleus is close to the radii of the Hoyle state in 12C.

Do analogs of the Hoyle state exist in 16O?

The root-mean-square radii of short-lived 0+-states in 16O were obtained for the first time from an analysis of α+16O diffraction scattering. Data on elastic and inelastic α+16O scattering were analyzed on the basis of the modified diffraction model in the range of projectile energies between a few tens of MeV units to 400 MeV. No case of a significant enhancement of the radius with respect to its ground-state value was observed. In particular, this concerns the 15.1-MeV 0 6 + state, which lies near the threshold for breakup to four alpha particles and for which we did not confirm a giant radius predicted by the alpha-particle-condensate model. This result disproves the hypothesis that the 16O nucleus in the 0 6 + state has a rarefied structure and appears to be an analog of the known Hoyle state at 7.65 MeV (0 2 + ) in 12C.

Proton halo in the 13N nucleus

We demonstrate that the radii of excited nuclear states can be estimated using the (3He, t) charge-exchange reaction and relying on the modified diffraction model. The radius of the N excited state with an excitation energy of E*=2.73 MeV, which lies in a continuous spectrum, is determined. The radius of this state proves to be close to that of the mirror 3.09-MeV state of the 13С nucleus, which possesses a neutron halo but lies in a discrete spectrum. Thereby, we demonstrate that the 2.37-MeV state of the 13N nucleus has a proton halo. The analysis is based on published measurements of differential cross sections for relevant reactions.

Rotational bands in 11B and identification of diluted states

Differential cross-sections of the 11B + α inelastic scattering at E(α) = 65 leading to the most of the known 11B states at the excitation energies up to 14 MeV were measured. The data analysis was done by DWBA and in some cases by the modified diffraction model allowing determining the radii of the excited states. The radii of the states with excitation energies less than ∼ 7 MeV with the accuracy not less than 0.1-0.15 fm coincide with the radius of the ground state. This result is consistent with the traditional view of the shell structure of the low-lying states in 11B. Most of the observed high-energy excited states are distributed among four rotational bands. The moments of inertia of band states are close to the moment of inertia of the Hoyle state of 12C. The calculated radii, related to these bands, are 0.7 - 1.0 fm larger than the radius of the ground state, and are close to the radius of the Hoyle state. These results are in agreement with existing predictions about various cluster structure of 11B at high excitation energies. The state with the excitation energy 12.56 MeV, Iπ = 1/2+, T = 1/2 and the root mean square radius R ∼ 6 fm predicted in the frame of the alpha condensate hypothesis was not found.

Cluster rotational bands in11B

Differential cross-sections of 11 B+α inelastic scattering at E (α ) =65 MeV leading to most of the known 11 B states at excitation energies up to 14 MeV were measured [1]. The data analysis was done using Modified diffraction model (MDM) [2] allowing determining radii of excited states. Radii of the states with excitation energies less than ∼ 7 MeV coincide with the radius of the ground state with an accuracy not less than 0.1 - 0.15 fm. This result is consistent with traditional view on shell structure of low-lying states in 11 B. Most of the observed high-energy excited states are distributed among four rotational bands. Moments of inertia of band states are close to the moment of inertia of the Hoyle state of 12 C. The calculated radii, related to these bands, are 0.7 - 1.0 fm larger than the radius of the ground state, and are close to the Hoyle state radius. These results are in agreement with existing predictions about various cluster structure of 11 B at high excitation energies.

States of13C with abnormal radii

Differential cross-sections of the elastic and inelastic 13 C + α scattering were measured at E(α ) = 90 MeV. The root mean-square radii( >) of 13 C nucleus in the states: 8.86 (1/2− ), 3.09 (1/2+ ) and 9.90 (3/2− ) MeV were determined by the Modified diffraction model (MDM). The radii of the first two levels are enhanced compared to that of the ground state of 13 C, confirming the suggestion that the 8.86 MeV state is an analogue of the Hoyle state in 12 C and the 3.09 MeV state has a neutron halo. Some indications to the abnormally small size of the 9.90 MeV state were obtained.

Neutron halo in the exotic first excited state of 9Be

Differential cross sections for elastic and inelastic 9Be + α scattering (the inelastic scattering leading to the 1.68- and 2.43-MeV states) were measured at the energies of 40 and 90 MeV. The radii of the excited states in question were determined on the basis of a modified diffraction model. The root-mean-square radius of the level at 1.68 MeV proved to be approximately 1 fm larger than the radii of both neighboring states. The reason is that, for this level, there is a neutron halo whose properties are similar to the properties of the known halo in the 11Be nucleus. This result is the first observation of a halo in an unbound state lying above the neutronemission threshold.

Analogues of the exotic Hoyle state in the 12C nucleus

Experimental data on inelastic scattering of alpha particles by the 11B, 12C, and 13C nuclei are analyzed using the modified diffraction model and the radii of these nuclei in some “abnormal” excited states are found. It is shown that the 0 2 + (7.65 MeV) Hoyle state in the 12C nucleus is the base state for a new 0 2 + –2 2 + –4 2 + rotational band (in addition to the ground-state band), in which the third member is the discovered 4 2 + (13.75 MeV) state. The radii of the 12C nucleus in the above-mentioned three states are 25–30% larger than its ground state radius. It is found that the radii of the 1/2– (8.86 MeV) state in the 13C nucleus and the 3/2– (8.56 MeV) state in the 11B nucleus are close to the radius of the Hoyle state in 12C and that a similar rotational band is based on the 8.56 MeV state. The above 13C and 11B states can be regarded as analogues of the Hoyle state. The prediction of the alpha-condensation model that a similar analogue in 11B is the 12.56 MeV state with a radius that is comparable with the nuclear radius of uranium was not confirmed.

Possible observation of an excited state with an anomalously small radius in the 13C nucleus

Differential cross sections for 13C(α, α') inelastic scattering at E(α) = 65 and 90 MeV with the excitation of the 9.90-MeV state in the 13C nucleus have been studied. The analysis within the modified diffraction model has shown that the rms radius of the nucleus in this state can be approximately 20% smaller than its radius in the ground state.

Spectroscopy of exotic states of13C

The differential cross-sections of the elastic and inelastic 13 C + α scattering were measured at E (α) = 65 MeV. The radii of the states: 8.86 (1/2¯), 3.09 (1/2 + ) and 9.90 (3/2¯) MeV were determined by the Modified diffraction model (MDM). The radii of the first two levels are enhanced relatively that of the ground state of 13 C, confirming the suggestion that the 8.86 MeV state could be an analogue of the Hoyle state in 12 C and the 3.09 MeV state has a neutron halo. No enhancement of the radius of the 9.90 MeV state was observed.

Cluster states in11B

The differential cross-sections of the elastic and inelastic 11 B + α scattering was measured at E(α) = 65 MeV. The analysis of the data by Modified diffraction model (MDM) showed that the RMS radii of the 11 B state 3/2 - , E* = 8.56 MeV is ~ 0.6 fm larger than that of the ground state. The 12.56 MeV state was not observed contrary to the predictions of the α-condensate model. The 13.1 MeV state was excited with the angular momentum transfer L = 4 confirming its belonging to the rotational band with the 8.56 MeV state as a head.

Radius of 12C in the excited 2 2 + Hoyle state

The differential cross sections of the inelastic α + 12C scattering leading to the excitation of the short-lived 2 2 + (E x = 9.84 or 9.6 MeV) state in 12C have been analysed within a modified diffraction model. We determined the diffraction radii of 12C in this excited state at E α = 386 and 240 MeV and compared them with the diffraction radius for the 0 2 + (E x = 7.65 MeV) Hoyle state. We found that the rms radii for the 2 2 + state is 〈R(2 2 + )〉 = 3.07 ± 0.13fm, which agrees well with the rms radius of the 0 2 + , 7.65 MeV Hoyle state and is a factor of 1.3 larger than the rms radius for the ground state of 12C. The similarity between the rms radii of the 0 2 + and 2 2 + states provides a strong argument in favor of a hypothesis that the 2 2 + state is the first member of a rotational band based on the 0 2 + Hoyle state.

Effect of neutron halos on excited states of nuclei

The differential cross sections of the inelastic scattering leading to the excitation of short-lived states in the stable ${}^{13}$C and ${}^{9}$Be nuclei as well as the radioactive ${}^{11}$Be nucleus have been analyzed. Signatures of neutron halos in the excited states located close to the neutron emission thresholds have been investigated by applying a recently developed modified diffraction model. The abnormally large rms radius was identified for the 3.089-MeV $1/{2}^{+}$ state of ${}^{13}$C. Significantly enlarged diffraction radii were found for the 1.68-MeV $1/{2}^{+}$ and the 3.05-MeV $5/{2}^{+}$ states of ${}^{9}$Be. The analysis of the diffraction radii of the weakly bound radioactive ${}^{11}$Be nuclei have shown that the 1.78-MeV $5/{2}^{+}$ and 3.41-MeV $3/{2}^{+}$ states in ${}^{11}$Be have the same diffraction radii as the $1/{2}^{+}$ ground state, which is known as the best example of a state with one-neutron halo.

Observation of abnormally large radii of nuclei in excited states in the vicinity of neutron thresholds

Differential cross sections for inelastic scattering leading to the excitation of some nuclear states situated near neutron-emission thresholds were analyzed. With the aid of a modified diffraction model, abnormally large radii were found for the 1/21+ state of the 13C nucleus at 3.09 MeV, for the first levels of positive-parity rotational bands in the 9Be (1/2+ level at 1.68 MeV and 5/2+ level at 3.05 MeV) and 11Be (5/2+ level at 1.78 MeV and 3/2+ level at 3.41 MeV) nuclei, and for the 21+ state of the 14Be nucleus at 1.54 MeV and 11− state of the 12Be nucleus at 2.7 MeV. All of these states possess signatures typical of neutron halos.

Analysis of thin-film solid solutions on single-crystal silicon by simulation of x-ray rocking curves: B-Si and Ge-Si binary alloys

B-Si and Ge-Si thin-film solid solutions on silicon at different compositions are analyzed by multiple-crystal x-ray diffraction. Computer simulations of the rocking curves are made with a diffraction model which, for dilute alloys, does not differ from that commonly reported in the literature. For sufficiently high solute atomic fractions, modifications are introduced to the Fourier coefficients of the crystal polarizability and to the parameters depending on the lattice strain. In the B-Si case, the comparison between strain and carrier profiles resulting from the simulations and electrical measurements, respectively, gives information on the fraction of substitutional B, the presence of precipitates, and their coherent or incoherent nature. In the Ge-Si case, the solute fraction, its depth gradient, and the static atomic disorder in the alloy are determined by means of the modified diffraction model. In particular, the determination of the atomic displacements around the lattice points in fully strained thin-film Ge-Si alloys give results larger than those evaluated by Monte Carlo calculations for relaxed solid solutions.