Elementary Methods for Solving Diophantine Equations

U ovom radu proučit ćemo elementarne metode za rješavanje diofantskih jednadžbi. Prvi dio rada bazira se na linearnim diofantskim jednadžbama i Euklidovom algoritmu i njegovoj važnosti u rješavanju ovog tipa jednadžbi. Osim toga, u radu su opisane metode za rješavanje nelinearnih diofantskih jednadžbi i Pitagorina jednadžba. Sve navedene metode rješavanja demonstrirane su na primjerima.In this paper we will study elementary methods for solving Diophantine equations. The first part of the paper is based on linear Diophantine equations, Euclidean algorithm and its importance in solving these types of equations. Furthermore, methods for solving nonlinear Diophantine equations and Pythagorean equation are described in the second part. Examples are used to demonstrate each method

Structure of 24Mg excited states and their influence on nucleosynthesis

Još od prvih proučavanja sustava 12C+12C, gdje su uočene izražene rezonancije u udarnom presjeku mjerenja elastičnog raspršenja te reakcijama prijenosa, ovaj sustav i dalje privlači pažnju. Posebno zato što reakcija 12C+12C, ima vrlo važnu ulogu u zvjezdanim sustavima bogatim ugljikom, kao što su super AGB zvijezde, supernove tipa Ia te superbljeskovi. Astrofizički značajno područje energije ovog sustava, za ove zvjezdane objekte, je između 1.5 i 3.3 MeV u energiji u sustavu centra mase sistema 12C+12C, što je ekvivalentno području između 15.4 i 17.2 MeV u energiji pobuđenja jezgre 24Mg. Dosadašnja mjerenja udarnog presjeka ove reakcije pokrivaju područje do 2.10 MeV. Kako se ovo područje energija od astrofizičke važnosti, nalazi ispod kulonske barijere, očekuje se da su samo stanja niskog spina, 0+ ili 1−, važna za proces gorenja ugljikom. Kod stanja višeg spina, postaje značajan i doprinos centrifugalne barijere, što čini udarni presjek još manjim. Mjerenje udarnog presjeka reakcije 12C+12C, u astrofizičkom području energija, je vrlo zahtjevno i dugotrajno. Komplementarni pristup ovom problemu je korištenje indirektnih tehnika mjerenja. U ovom radu su predstavljene dvije indirektne tehnike mjerenja, metoda rezonantne čestične spektroskopije pomoću koje je proučavana reakcija 16O(12C,α)24Mg*, te metoda rezonantnog elastičnog raspršenja za mjerenje reakcije 4He(20Ne,4He)20Ne. Glavna ideja ovog rada je proučavanje rezonantnih struktura jezgre 24Mg, s posebnim naglaskom na stanja opažene u astrofizički značajnom području energija, spina 0+ ili 1−. Analiza rezonancija opaženih u različitim kanalima raspada 24Mg, te izračunati omjeri grananja tih stanja, su pomogli u razumijevanju karakteristika niza pobuđenih stanja 24Mg. Detaljnom analizom je utvrđeno, da u ova dva mjerenja, nije opaženo stanje spina 0+ ili 1− u astrofizički značajnom području energija. No ne može se isključiti mogućnost postojanja takvog stanja, koje nije moglo biti opaženo radi prevelike gustoće stanja te nedovoljno dobre rezolucije mjerenja. Analiza predstavljena u ovom radu, ukazuje na probleme koji postoje u traženju astrofizički značajnog stanja, primjenom indirektnih metoda rezonantne čestične spektroskopije i rezonantnog elastičnog raspršenja. Dobiveni rezultati mjerenja metodom rezonantne čestične spektroskopije su ukazali na značajan doprinos strukture u kojoj je 20Ne u svojim pobuđenim stanjima, za stanja 24Mg koja se raspadaju u kanal α+20Ne. Ta informacija je značajno olakšala analizu mjerenja metodom rezonantnog elastičnog raspršenja. Analizom mjerenja izvršenog metodom rezonantnog elastičnog raspršenja, određeni su parametri više pobuđenih stanja 24Mg, a koji do sad, nisu bili dostatno poznati.Since the first study of the 12C+12C system, which showed prominent resonances in the elastic scattering and the transfer reaction cross section measurements, study of this system is relevant even today. Especially because of the important role that the 12C+12C reaction has on the carbon-rich stellar systems, such as super AGB stars, supernovae Type Ia and superbursts. The relevant energy range of the cross section measurements for these stellar systems is between 1.5 and 3.3 MeV in the energy of the center of the mass of the 12C+12C system, which is equivalent to 15.4-17.2 MeV in the excitation energy of 24Mg. Current cross section measurements of the 12C+12C system extend down to 2.10 MeV. Since this region of astrophysical interest is far below Coulomb barrier, it is expected that only 0+ or 1− state is relevant for the carbon burning process. The cross section measurement of the 12C+12C reaction, in the astrophysical region of interest, is difficult and long lasting. Different approach to this problem is made by using indirect techniques of measurement. In this work, two indirect techniques of measurement are presented, the resonant particle spectroscopy method that studied 16O(12C,α)24Mg* reaction, and the resonant elastic scattering method, that studied the 4He(20Ne,4He)20Ne reaction. The main idea of this work was to study the resonance structures of 24Mg, with strong emphasis on the resonances observed in the astrophysical region of interest, with spin 0+ or 1−. The analysis of the resonances observed in different decay channels, their branching ratios, has been performed in order to understand the structure of 24Mg excited states. Detailed analysis showed that in both measurements, the 0+ or 1− state in the astrophysical region of interest, was not observed. Although one can not exclude the existence of that kind of state, that could not be observed due to the high density of the states and not high enough resolution of the experiments. Analysis presented in this work, showed the problems that one experiences by using indirect methods of measurement, the resonant particle spectroscopy method and the resonant elastic scattering method, to observe the astrophysical relevant state. The results of the analysis of the resonant particle spectroscopy method measurement, showed high contribution of the structures, in which 20Ne is in its excited states, for the states in 24Mg that decay into the α+20Ne channel. This information made the analysis of the measurement done by the resonant elastic scattering method much easier. The analysis of the resonant elastic scattering method measurement determined parameters of many of the 24Mg excited states, that weren’t well known before

Study of interstrip gap effects and efficiency for full energy detection of Double Sided Silicon Strip Detectors

In this work is reported a study on the response of double sided silicon strip detectors. In order to investigate the effect of the electrode segmentation on the detector response, two experiments were performed aimed to measure the efficiency for full energy detection. Results show that the efficiency for full energy detection, that is directly related to effective width of the inter-strip region, varies with both detected ion energy and bias voltage. The experimental results are qualitatively reproduced by a simplified model based on the Shockley-Ramo-Gunn framework

The identification of α -clustered doorway states in 44, 48, 52 Ti using machine learning

Abstract: A novel experimental analysis method has been developed, making use of the continuous wavelet transform and machine learning to rapidly identify α-clustering in nuclei in regions of high nuclear state density. This technique was applied to resonant scattering measurements of the 4He(40,44,48Ca,α) resonant reactions, allowing the α-cluster structure of 44,48,52Ti to be investigated. Fragmented α-clustering was identified in 44Ti and 52Ti, while the results for 48Ti were less conclusive, but suggest no such clustering

Mass correlation between light and heavy reaction products in multinucleon transfer 197Au+130Te collisions

We studied multinucleon transfer reactions in the 197Au+130Te system at Elab=1.07 GeV by employing the PRISMA magnetic spectrometer coupled to a coincident detector. For each light fragment we constructed, in coincidence, the distribution in mass of the heavy partner of the reaction. With a Monte Carlo method, starting from the binary character of the reaction, we simulated the de-excitation process of the produced heavy fragments to be able to understand their final mass distribution. The total cross sections for pure neutron transfer channels have also been extracted and compared with calculations performed with the grazing code

Light and heavy fragments mass correlation in the 197Au+130Te transfer reaction

We studied multinucleon transfer (MNT) processes in the 197Au+130Te at Elab=1.07 GeV system coupling the PRISMA magnetic spectrometer to NOSE, an ancillary particle detector. We constructed a mass correlation matrix associating to each light fragment identified in PRISMA the corresponding mass distribution of the heavy partner detected in NOSE and, through the comparison with Monte Carlo simulations, we could infer about the role of neutron evaporation in multinucleon transfer reactions for the population of neutron-rich heavy nuclei

Structure of 24Mg excited states and their influence on nucleosynthesis

The main idea of the two presented experiments is to study the decay of resonances in 24Mg at excitation energies above the 12C+12C decay thresh- old, in the astrophysical energy region of interest. The measurement of the 12C(16O,α)24Mg* reaction was performed at INFN-LNS in Catania. Only the α+20Ne decay channel of 24Mg is presented here, because it was a motivation for conducting a new experiment, a study of the 4He(20Ne,4He)20Ne reaction, performed at INFN-LNL in Legnaro. Some preliminary results of this measurement are also presented

The thick target inverse kinematics technique with a large acceptance silicon detector array

An experimental technique for studying elastic scattering using a thick gas target is described, with a measurement of the α(24Ne, α) reaction used as an example. Advantages such as ease, detector efficiency, and the possibility of measuring the cross section at 180° in the centre-of-mass are discussed. It is shown that a resolution of tens of keV is practical at zero degrees, and that the dominant contribution to the resolution for large angles is angular straggling of the beam in the entrance window. The use of helium gas as the target allows direct measurement of a-cluster states