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Geometrija baždarnih teorija
No full textThe goal of this thesis was to mathematically explain concepts from gauge theories. The most emphasis was put on explaining gauge transformations, covariant derivatives, and curvature. The main tools we used for such description were principal fiber bundles and connections. As there exists a lot of background geometry, preparations had to be made before the introduction of those two. Those preparations included an understanding of fiber bundles and Lie groups. Fiber bundles allowed for the description of spaces are not trivial, i.e. those that can be described only locally. Lie groups, on the other hand, were used as a tool for the description of the symmetries we were dealing with. On principal fiber bundles connections were defined as a formal way of comparison that is not well defined in Euclidean geometry. This helped us understand gauge transformations. However, for a full description of covariant derivatives, associated bundles need to be mentioned as they serve as a tool for defining parallel transport map in the context of a vector bundle in which we know how to compare points. This is enough for a formal definition of a covariant derivative, which can be found in both particle physics and general relativity. Curvature relied on no more new concepts as understanding covariant derivative and exterior covariant differentiation was enough for its description. Examples of curvature in gauge theories were given alongside terminology used in these fields as naming conventions differ
PUP HrZZ projecta "Astronomija gama-zraka visokih energija u doba višečestičnih opažanja" (IP-2022-10-4595)
No full textMUonE experiment: Muon indentification
No full textUnutar okvira MUonE projekta razvoja novog detektora za muone u CERNu, pronađena je potreba za novom vrstom detektora zbog starosti i gubitka tehničke ekspertize u održavanju i proizvodnji trenutačnih eksprimentalnih postava. Predložen je i napravljen prototip detektora muona na bazi scintilatorskih vlakana i Silicone Photo Multiplier (SiPM) fotoosjetljivih detektora. SiPMovi su se sastojali od rešetke Single Photon Avalanche Diode (SPAD) elemenata sa brzim i sporim izlazom. U radu smo se fokusirali na kalibraciju sporog izlaza. Scintilacijska vlakna emitiraju fotone u kontaktu sa zrakom nabijenih čestica, poput elektrona i muona, te iste provode do fotoosjetljivog detektora sa visokom stopom efikasnosti. Za prikupljanje podataka se koristila razvojna pločica Citiroc 1A, proizvođača Weerock. Kučište koje drži pločicu sa SiPM čipovima i scintilacijska vlakna je 3D modelirano i printano na temelju mjera pločice. U sklopu rada je rad dvanaest SiPM čipova kalibriran, te su ispitane karakteristike rada u mraku i pod utjecajem simulacije izvora u obliku pulsirajuće plave LED, sa i bez scintilacijskih vlakana. Napravljene su staircase kalibracijske krivulje u ovisnosti o naponu reverzne polarizacije SiPM čipova i S-curve kalibracijske krivulje u ovisnosti o frekvenciji sata mjerenja, napona na LED i duljini impulsa struje kroz LED. Zaključeno je da je idealno postaviti prag detekcije signala na 3.5 fotoelektrona, odnosno 400 DACu, kako bi se u potpunosti isključio šum elektronike, cross-talk efekt između različitih SPAD elemenata jednog SiPM čipa, te lažni signal zbog curenja svjetlosti. Također je zaključeno da scintilacijska vlakna značajno pospješuju detekciju signala.Within CERN, there is an initiative for the development of a new muon detector under the name MuonE. It has determined a need for a new detection technology because of the age and loss of expertise in the maintaining and production of the existing experimental setups. A prototype muon detector based on scintillating fibers and Silicone Photo Multiplier (SiPM) chips has been proposed and made. SiPMs consist of an array of Single Photon Avalanche Diodes (SPAD) with both slow and fast outputs. Scintillation fibers emit photons in contact with a beam of charged particles, such as electrons and muons. They also act as an optical fiber and conduct said photons to photo sensitive detectors with a high degree of efficiency. Development board Citiroc 1A by the company Weerock was used for data acquisition. The housing of the SiPM board and scintillating fibers was modeled and 3D printed based on the measurements of said board. In the scope of this work, twelve SiPM detection channels have been calibrated. Their work has been characterised in both darkness and under the influence of a source simulation, with and without scintillating fibers. A blue LED was used as the source simulation. Staircase calibration curves were made in the response to the reverse polarisation potential of the SiPM chips, as well as S-curve calibration curves in the response to the measurement clock frequency, LED voltage and the duration of the current impulse through the LED. It was concluded that the threshold of detection should be set at 3.5 electronvolts, which is 400DACu. This completely excludes electrical noise, cross-talk effect between the different SPADs of one SiPM chip, and false positives due to light leakages. It has also been determined that the scintillating fibers significantly improve signal detection
Feynman's path integrals
No full textU radu istražujemo formalizam Feynmanovih integrala po putevima kao alternativni pristup
kvantnoj mehanici (QM). Integral po putevima povezuje klasičnu i kvantnu teoriju
kroz princip minimalne akcije. U uvodnom dijelu, pomoću eksperimenta dvostrukog proreza,
prikazujemo osnovne kvantne principe kao npr. superpozicija, te uvodimo koncept
kvantnih prijelaza kao sumu preko svih mogućih puteva. Nadalje, prikazujemo kako se formalizam
integrala po putevima izvodi iz kanonske QM. Bavimo se tehnikama računanja
integrala po putevima, detaljno analizirajući Gaussove integrale koji pomažu u izračunavanju
propagatora. Dalje, fokusiramo se na konkretne primjere primjene integrala po
putevima na fizikalne sisteme poput harmoničkog oscilatora, detaljno prikazujući izračune
za propagatore općenitijih lagranžijana. Konačno, istražujemo vezu između integrala po
putevima i statističke fizike, objašnjavajući kako se particijske funkcije mogu izračunati
pomoću ovog formalizma.In this thesis, we explore the formalism of Feynman’s path integrals as an alternative
approach to quantum mechanics (QM). The path integral connects classical and quantum
theory through the principle of least action. In the introductory section, we illustrate
fundamental quantum principles such as superposition using the double-slit experiment,
and introduce the concept of quantum transitions as a sum over all possible paths. Furthermore,
we demonstrate how the path integral formalism can be derived from canonical
QM. We delve into techniques for computing path integrals, analyzing Gaussian integrals
that aid in the calculation of propagators. We then focus on specific applications of path
integrals to physical systems such as the harmonic oscillator, providing detailed calculations
for propagators of more general Lagrangians. Finally, we investigate the connection
between path integrals and statistical physics, explaining how partition functions can be
computed using this formalism
The measurements of the solar differential rotation from the Kanzelhöhe sunspot drawings for thee solar cycle No. 19
No full textSunčev dinamo odgovoran je za kompleksnu magnetsku strukturu Sunca. Jedan od ključnih elemenata za održavanje Sunčevog dinama je diferencijalna rotacija Sunca, čija se mjerenja u principu zasnivaju na tri metode: heliosizmologija, spektroskopija i praćenje pomaka ustrojstava. Sunčeve pjege imaju prednost zbog dostupnih dugoročnih opažanja i zapisa podataka u usporedbi s ostalim ustrojstvima. Mnogi opservatoriji čuvaju baze podataka s crtežima pjega na Sunčevom disku, a među onima s dugom tradicijom nalazi se i Kanzelhöhe Observatory for Solar and Enviromental Research (KSO). U ovom radu analizirana je diferencijalna rotacija Sunca koristeći Kanzelhöhe slike Sunčevog diska za 19. ciklus Sunčeve aktivnosti (1954 - 1964) s fokusom na određivanje parametara (A, B) diferncijalne rotacije i određivanje asimetrije u rotaciji Sunca oko ekvatora (asimetrija sjever
- jug). Realizacija ovog cilja ostvarena je jednostavnim postupkom koji se sastoji od četiri koraka. Na početku su manualno identificrani položaji Sunčevih pjega i grupa pjega. Sinodičke brzine izračunate suiz položaja grupe pjega s dvije različite metode: metodom dnevnog pomaka (DS metoda) grupe pjega i robustnom metodom najmanjih kvadrata (rLSQ metoda), a sinodička rotacijska brzina pretvorena je u sideričku rotacijsku brzinu koristeći specifični korekcijski faktor koji odgovara relativnom gibanju Zemlje i Sunca u danom trenutku. U konačnici su određeni parametri diferencijalne rotacije A i B prilagodbom izraza ω = A + B sin^2(b) na sideričke brzine ω u ovisnosti o pridruženim kvadratima sinusa heliografskih šrina sin^2(b). Kvalitativni opis Sunčeve rotacija za 19. ciklus Sunčeve aktivnosti (1954 - 1964) u skladu je sa standardnim modelom diferencijalne rotacije Sunca. Konkretno rečeno, kutna brzina rotacije Sunca smanjuje se od ekvatora prema polovima na simetričan način za obje hemisfere
(N i S), sa specifičnim kvantitativnim profilom:
ωsid = (14.39 ± 0.02) ◦/dan − (2.70 ± 0.15) sin^2(b) ◦/dan
ωsid = (14.33 ± 0.02) ◦/dan − (2.31 ± 0.12) sin2(b) ◦/dan
za rLSQ i DS metodu, respektivno.The solar dynamo is responsible for the tangled magnetic structure of the Sun. One of the key elements for maintaining the Sun’s dynamo is the Sun’s differential rotation, which is measured using three principal methods: helioseismology, spectroscopy, and tracking the movement of solar structures (the tracer method). Sunspots have the advantage of long-term observations and data records compared to other tracers. Many observatories retain databases with patterns of sunspots on the solar disc, and among those with a long tradition is the Kanzelhöhe Observatory for Solar and Environmental Research (KSO). In this paper, the differential rotation of the Sun is analyzed using Kanzelhöhe drawings of the solar disc for the solar cycle No. 19 (years 1954 - 1964), with a focus on determining the parameters (A, B) of the differential rotation and evaluating the asymmetry in the Sun’s rotation around the equator. The realization of this aim was achieved through a simple procedure consisting of four steps. In the first step, the positions of sunspots and sunspot groups were identified manually. The synodic velocities were calculated from the positions of the sunspot groups using two different methods: the daily shift method (DS method) and the robust least squares method (rLSQ method). The synodic rotation velocity was then converted to the sidereal rotation velocity using a specific correction factor that accounts for the relative motion of theEarth and the Sun at a given moment. At the end, the parameters of differential rotation, A and B, were determined by fitting the expression ω = A + B sin^2 (b) to the sidereal velocity ω depending on the corresponding squares of the sine of the heliographic latitude sin^2(b). The qualitative description of the Sun’s rotation for the solar cycle No. 19 (years 1954 - 1964) is consistent with the standard model of differential rotation. In particular, the angular velocity of the Sun’s rotation decreases from the equator toward the poles, symmetrically in both hemispheres (N and S), with a specific quantitative profile of rotation:
ωsid = (14.39 ± 0.02) ◦/dan − (2.70 ± 0.15) sin^2(b) ◦ /dan
ωsid = (14.33 ± 0.02) ◦/dan − (2.31 ± 0.12) sin^2(b) ◦ /dan
for the rLSQ and DS method, respectively
Determination of distance and stellar parameters by Bayes probabilistic inference
No full textThis thesis is devoted to the study of the stellar parameters and determination of stellar distances. For that purpose, the BEAST (The Bayesian Extinction and Stellar Tool) algorithm was used. BEAST is fitting ultraviolet to near-infrared photometric magnitudes to precomputed model grids. This method precisely evaluates parameters such as age, mass, metallicity, and distance by implementing Bayesian inference combined with prior distributions best-suited to the characteristics of the stellar population under study. BEAST integrates synthetic photometry and uses chi-squared fitting techniques that handle observational data from large surveys. This study was focused on globular clusters M92, M13 and NGC 6791 and with the help of BEAST the following parameters were determined: mass, luminosity, surface gravity, radius, temperature, distance, age and metallicity of the stars we studied. We also determined dust parameters: visual extinction and ratio of total to selective extinction. We later compared values from other science papers and values obtained by the BEAST. For M92, BEAST determined the mean distance to be 5.46 kpc (accuracy of 66.6%). For age, BEAST determinated the value of 7.45 Gyr (accuracy of 67.7%). For visual extinction in magnitude, BEAST got Av = 0.12 (accuracy of 47.3%) and for ratio of total to selective extinction, Rv = 2.76 (accuracy of 89%). For M13, BEAST BEAST determinated mean distance to be 5.01 kpc (accuracy of 63.67%). For age, it determinated 7.67 Gyr (accuracy of 65.84%). For visual extinction in magnitude BEAST got Av = 0.16 (accuracy of 87.5%) and for ratio of total to selective extinction, Rv = 3.49 (accuracy of 88.83%). For NGC 6791, BEAST calculated mean distance to be equal to 4.55 kpc (accuracy of 92.98%), which is an excellent result. In that cluster we analysed SGB and RGB stars, while in other clusters we analysed AGB, RGB and HB stars, so it might be the case that BEAST is doing better estimates for stars in earlier stages of their evolution. For age, BEAST estimated the value of 7.77 Gyr (accuracy of 93.61%). For visual extinction in magnitude we obtained Av = 0.46 (accuracy of 65.71%) and for ratio of total to selective extinction, Rv = 3.01 (accuracy of 97.1%). We have also shown that fixing stellar age does not influence the determination of other stellar and dust parameters, while fixing distance can greatly influence stellar luminosity and radii. BEAST is very effective in dealing with the challenges posed by interstellar dust extinction, which can greatly affect the accuracy of photometric data. The results are not perfect, but they show the effectiveness of Bayesian methods in dealing with complex data in astronomy
Formation and Evolution of Protoplanetary Disks and Protoplanets
No full textU ovom radu bavimo se nastankom i razvojem protoplanetarnih diskova te procesima koji dovode do formiranja protoplaneta i planetarnih sustava. Protoplanetarni diskovi su rotirajući diskovi plina i prašine koji se formiraju oko mladih zvijezda uslijed gravitacijskog kolapsa molekularnih oblaka. U njima, akrecijom materijala nastaju planetezimali, protoplaneti i konačno planeti. Opisujemo strukturu protoplanetarnih diskova, uključujući radijalnu i vertikalnu raspodjelu temperature i materijala, te razvoj diskova kroz faze do konačnog raspršenja plina i nastanka planetarnog sustava. Također, detaljnije su razmatrani mehanizmi migracije protoplaneta, koagulacija prašine i akrecija, te njihova uloga u nastanku različitih tipova planeta poput plinovitih divova i terestričkih planeta. Rad pruža pregled aktualnih teorijskih modela i opažanja, naglašavajući njihovu važnost u razumijevanju planetarnih sustava.In this thesis, we describe the formation and evolution of protoplanetary disks and the processes that lead to the formation of protoplanets and planetary systems. Protoplanetary disks are rotating disks of gas and dust that form around young stars as a result of the gravitational collapse of molecular clouds. Within these disks, the accretion of material leads to the creation of planetesimals, protoplanets, and ultimately planets. We describe the structure of protoplanetary disks, including the radial and vertical distribution of temperature and materials, and the evolution of disks through stages until the final dispersal of gas and the formation of planetary systems. Additionally, we discuss in more detail the mechanisms of protoplanet migration, dust coagulation, and accretion, and their roles in the formation of different types of planets, such as gas giants and terrestrial planets. The thesis provides an overview of current theoretical models and observations, emphasizing their importance in understanding planetary systems
Investigation of corrosion behaviour of low-alloy steel
No full textU ovom diplomskom radu teorijski su obrađeni osnovni pojmovi i mehanizmi korozije, s posebnim naglaskom na elektrokemijsku koroziju. Ispitivana je otpornost čelika 42CrMo4 na koroziju u različitim medijima primjenom elektrokemijskih metoda i pretražnog elektronskog mikroskopa (SEM). Ispitivanja su provedena u vodovodnoj vodi, 3,5%-tnoj otopini natrijevog klorida, 0,1 M otopini sumporne kiseline i 0,1 M otopini dušične kiseline. Rezultati su pokazali da se brzina korozije značajno razlikuje ovisno o mediju, pri čemu je najveća brzina korozije zabilježena u sumpornoj kiselini, a najmanja u vodovodnoj vodi. SEM/EDS analiza otkrila je različite stupnjeve korozijskih oštećenja na površini uzoraka, s najizraženijim promjenama u kiselim medijima. Na temelju dobivenih rezultata, zaključeno je da čelik 42CrMo4 iskazuje visoku osjetljivost na kisele uvjete, dok u manje agresivnim medijima pokazuje veću otpornost na koroziju.This master thesis provides a theoretical overview of the basic concepts and mechanisms of corrosion, with a particular emphasis on electrochemical corrosion. The study investigates the corrosion resistance of 42CrMo4 steel in various media using electrochemical methods and scanning electron microscop (SEM). The experiments were conducted in tap water, 3.5% sodium chloride solution, 0.1 M sulfuric acid solution, and 0.1 M nitric acid solution. The results indicated that the corrosion rate of 42CrMo4 steel varies significantly depending on the medium, with the highest corrosion rate recorded in sulfuric acid and the lowest in tap water. SEM/EDS analysis revealed different degrees and forms of corrosion damage on the sample surfaces, with the most pronounced changes observed in acidic media. Based on the results, it was concluded that 42CrMo4 steel exhibits high sensitivity to acidic conditions, while demonstrating better corrosion resistance in less aggressive environments
Identifikacija muona u MUonE eksperimentu
No full textThis thesis investigates the performance of the so called muon filter, a critical component of the MUonE experiment aimed at precisely determining the hadronic leadingorder (HLO) corrections to the muon’s anomalous magnetic moment. The filter, located
downstream of the electromagnetic calorimeter, is designed to identify muons and reduce
hadronic contamination, particularly from pions. Using simulations conducted with FairMUonE software on the Bura supercomputer, various absorber layer configurations were
analyzed to optimize the filter’s performance.
Key findings reveal that increasing absorber thickness does not impact the acceptance of the muon detector significant enough to compromise detection efficiency. The
analysis also demonstrated that the distinct angular distributions of reconstructed muons
and background particles enable the development of the Angle Method, which effectively
identifies muon tracks by applying a cut-off on angular distributions. This method shows
a success rate exceeding ∼ 99% in distinguishing muons from background noise. Furthermore, the simulations confirmed the essential role of the muon filter in enhancing particle
identification, especially in ambiguous regions where muon and electron angles are similar.
Without the muon filter, up to ∼ 80% of events in these regions could not be identified.
The results underscore the necessity of the muon filter for accurate measurements in
the upcoming 2025 test run, providing valuable insights for future experimental setups
and strategies
Spinors in classical mechanics
No full textSpinors were first mentioned by Elie Cartan (1913) and were introduced in physics by Dirac.
Though it is conventionally assumed spinor wave functions are a consequence of quantum weirdness
or relativity, and while on closer inspection they can be seen as having topological origin, it can be
shown that the fundamental reason for the appearance of spinorial objects in quantum mechanics
is purely geometrical. Spinors are naturally described in the language of Clifford algebra which
Clifford himself originally intended to be a universal geometric language. To that end, the Kepler
problem of classical mechanics is solved in a Newtonian geometric spirit albeit in a modern algebraic
formulation, setting the stage for the formulation of the spinor equation of motion which linearizes
and regularizes (removes singularity) the inverse square force equation of motion with universal
solutions yielding new applications to perturbation theory. This also leads to new insights and
interpretations of quantum mechanical observables, and the geometric nature of fermionic spin
1/2 wave functions.Spinors were first mentioned by Elie Cartan (1913) and were introduced in physics by Dirac.
Though it is conventionally assumed spinor wave functions are a consequence of quantum weirdness
or relativity, and while on closer inspection they can be seen as having topological origin, it can be
shown that the fundamental reason for the appearance of spinorial objects in quantum mechanics
is purely geometrical. Spinors are naturally described in the language of Clifford algebra which
Clifford himself originally intended to be a universal geometric language. To that end, the Kepler
problem of classical mechanics is solved in a Newtonian geometric spirit albeit in a modern algebraic
formulation, setting the stage for the formulation of the spinor equation of motion which linearizes
and regularizes (removes singularity) the inverse square force equation of motion with universal
solutions yielding new applications to perturbation theory. This also leads to new insights and
interpretations of quantum mechanical observables, and the geometric nature of fermionic spin
1/2 wave functions