High-precision (p,t) reactions to determine reaction rates of explosive stellar processes Matić, Andrija

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1 University of Groningen High-precision (p,t) reactions to determine reaction rates of explosive stellar processes Matić, Andrija IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 27 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Matić, A. (27). High-precision (p,t) reactions to determine reaction rates of explosive stellar processes. Groningen: s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to maximum. Download date:

2 High-precision (p,t) reactions to determine reaction rates of explosive stellar processes Andrija Matić

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4 RIJKSUNIVERSITEIT GRONINGEN High-precision (p,t) reactions to determine reaction rates of explosive stellar processes Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op maandag juni 27 om 6.5 uur door Andrija Matić geboren op 2 december 974 te Valjevo, Serbia

5 Promotor: Prof. dr. M.N. Harakeh Copromotor: Dr. A.M. van den Berg Beoordelingscommissie: Prof. dr. G. P. A. Berg Prof. dr. K. Hatanaka Prof. dr. C. E. Rolfs

6 Contents Introduction. Explosive stellar scenarios Supernovae type II Novae Supernovae type Ia X-ray bursts CNO cycles NeNa and MgAl cycles Astrophysical importance of 22 Na and 26 Al Na Al Outline of this work Theoretical model of stellar reaction rates 2. Stellar reaction rates Non-resonant reaction rates (direct reaction rates) Resonance reaction rates through narrow resonances Reactions through broad resonances The astrophysically relevant excitation-energy regions for 22 Mg and 26 Si Experimental setup and method Experimental area The GR spectrometer Matching between beam line and spectrometer Over-focus mode The detection and data-acquisition systems Experimental settings for the (p,t) reaction i

7 Contents 4 Calibration Software corrections Background subtraction Reference data Fits of the spectra Calibration function Calibration of 26 Si spectra Mg data and their astrophysical implications 6 5. The 24 Mg(p,t) 22 Mg angular distributions Mg and its mirror nucleus 22 Ne Calibration region (g.s MeV) Region above the proton-emission threshold (5.542 MeV MeV) Region above the α-emission threshold (8.42 MeV-.5 MeV) Region above.5 MeV Astrophysical implications for the 8 Ne(α,p) 2 Na reaction Astrophysical implications for the 2 Na(p,γ) 22 Mg reaction Summary Si data and their astrophysical implications Si and its mirror nucleus 26 Mg Calibration region (g.s MeV) Region above the proton-emission threshold (5.523 MeV MeV) Region above the α-emission threshold (9.64 MeV) Astrophysical implications for the 25 Al(p,γ) 26 Si reaction Astrophysical implications for the 22 Mg(α,p) 25 Al reaction Summary Summary and conclusion 23 A DWBA and CC calculations 27 Nederlandse samenvatting en vooruitblik 3 Acknowledgment 35 Bibliography 37 ii

8 Chapter Introduction. Explosive stellar scenarios One of the goals of nuclear astrophysics is to explain the energy production in stars, and related to this, creation and abundance of the chemical elements. Since the discovery in 952 by Merrill [] of radioactive technetium in spectra of S-stars, it became clear that nuclear synthesis of heavier nuclei in a star s interior does happen indeed. Furthermore, since technetium has a half-life of 5 to 6 years, which is a very short time period on a cosmological scale, this implies that nuclear synthesis also takes place in our present times. According to the Big-Bang theory matter created shortly after the Big Bang consists predominantly of H and 4 He. The heaviest created progenitor nucleus produced in significant abundance is 7 Li, whereas all heavier nuclei are created in stars; see Ref. [2]. The so-called main-sequence stars, which encompass the majority of stars, fuse hydrogen into helium. After the exhaustion of hydrogen in the core of a star, the core will contract because of the gravitational force and consequently the temperature and the pressure in the core will rise. Under these conditions the helium burning process will start and carbon (C), and oxygen (O), will be produced. After the exhaustion of helium in the core, its contraction will resume, leading to the ignition of C and O burning processes through which heavier elements are created. 22 Na and 26 Al are two long-living nuclei which eventually β + decay followed by subsequent γ-ray emission in the respective daughter nuclei. In case of explosive-burning scenarios, these nuclei can be ejected into the inter-stellar medium, and their subsequent γ-ray radiation can be detected with existing γ-ray observatories in outer space. Therefore, in this section we will discuss 4 explosive-burning scenarios for stars; these are novae, supernovae type I, supernovae type II, and X-ray bursts. Some nuclear processes which take place during the phases of explosive burning and which lead to the creation of heavier elements than C and O will be discussed as well. These explosive scenarios have to be taken into account in order to explain the isotopic abundances observed throughout the universe.

9 2. Introduction.. Supernovae type II A typical star for which one can apply the supernova II model has a mass of more than about 8 M (mass of the Sun). These stars evolve much faster than most main-sequence stars, which have typical lifetimes of about 8 years. During its lifetime a star goes through many burning stages, starting with the cycles of hydrogen burning up to silicon burning, subsequently leading to the formation of a Fe-Ni core. Because the iron nuclei are most tightly bound, a star cannot produce energy by the fusion of nuclei using iron or nickel as seed. Endothermic nuclear reactions will continue at extremely high temperatures of about 4 T 9 (T 9 = 9 K). These reactions are γ-ray photo-disintegration of nuclei and electron capture on protons leading to the production of neutrons and escaping neutrinos. The stellar iron core loses energy through these two reactions, and consequently the pressure of the degenerate electrons in the core decreases. Subsequently, the core contracts under the gravitational force and the temperature increases by compression. This dynamical collapse is starting at a density of about 9 g cm 3 and goes on to nearly nuclear matter density ( 4 g cm 3 ). At that moment, the density in the core can not be increased much further and the core collapse is stopped abruptly (in the order of a few milliseconds) leading to the collapsing material bouncing on the core. The bounce of the stellar material is very hard. Consequently, this leads to an outward moving compression wave which has such a high speed that it creates a shock wave. If the shock wave is strong enough material can get such a high energy, leading to a velocity beyond that of the escape velocity. In this case a large part of the stellar mass is ejected and this is the supernovae phenomenon. If the mass of the progenitor star is between 8-2 M, the stellar core will evolve to a neutron star. If the progenitor mass is above 2 M, the stellar core evolves to a black hole...2 Novae In case of a so-called nova, one is dealing with a binary system. One of the stars is a white dwarf and the other one is a star near the main sequence or an aging star such as red giant. The white dwarf can be a star mainly consisting of carbon-oxygen (CO) or oxygen-neonmagnesium (ONeMg), formed after the helium-burning (He) or carbon-burning stages, respectively. In low-mass main-sequence stars hot gaseous matter can be described by the ideal gas law. In these circumstances a velocity distribution of electrons and nuclei can be described by the Maxwell-Boltzmann velocity distribution. In case a star has consumed its nuclear fuel, it will collapse under the gravitational force with a huge increase in density. Under an enormous matter density the electron energy distribution is changed due to the Pauli exclusion principle. As the stellar material shrinks, its volume decreases and the number of states in a unit energy interval is reduced. Consequently, there will be less quantum states at

10 .. Explosive stellar scenarios 3 lower energies than is necessary for the Maxwell-Boltzmann distribution to apply and more electrons remain at higher energy than would be expected from Maxwell-Boltzmann. As the density continues to increase more and more electrons will remain with higher energy, up to the moment when almost all states up to the Fermi level are filled. At this moment the electrons form a degenerate gas which is under a huge pressure because of the electrons rapid motion. The high pressure provided by the electrons prevents further compression of the stellar material. In case of a white dwarf the electrons are pressed so tightly that further compression is not possible. The Roche lobe is a space around a star in a binary system which contains all material bound to that star. If a star expands beyond its Roche lobe, material can fall onto the other member of the binary system. In case of a nova the latter is a white dwarf and a hydrogen-rich accretion envelope can be created around it. This results from material falling from the star near the main sequence onto its accompanying star, the white dwarf. Since more and more hydrogen-rich material is accreted on top of the white dwarf, the temperature of this envelope will rise up to the moment that the hydrogen burning process starts. This temperature is around.2 T 9. Because electrons are in a degenerate state, gas material can not expand or cool before the degeneracy is lifted, and hence a rapid rise of temperature under constant pressure and density will occur. Degeneracy is lifted when the local temperature reaches the Fermi temperature, causing a rapid expansion of the envelope material into the interstellar medium. The typical nova peak temperature is T 9. During a nova explosion the burning of hydrogen proceeds through the CNO cycles (see section.2). Under these circumstances, high temperatures can be achieved and the CNO cycles can be broken, leading to rapid proton-capture (rp) reactions. These rp reactions can produce nuclei with a mass higher than those taking part in the CNO cycles...3 Supernovae type Ia The model of a supernova type Ia is a binary system of a CO white dwarf and a companion star. Various types of companion stars have been assumed in different models. In the most common scenario, the white dwarf accretes material from a companion star up to the moment that its core mass reaches the Chandrasekhar mass [3]. At that moment the gravitational force is strong enough to overcome the pressure from the degenerate electron gas, leading to a collapse of the star. At these high pressures and temperatures carbon and oxygen ignite in the core of the white dwarf and the burning front propagates outwards. During a supernova explosion, a sequence of runaway nuclear reactions occur in the core of the star. This is in contrast to regular nova explosions, where a thermonuclear runaway happens at the bottom of the accreted envelope. Furthermore, a supernova type Ia explosion releases a much greater amount of energy than a typical nova explosion. It is believed that a supernova type Ia is not an important contributor to nucleosynthesis beyond iron.

11 4. Introduction..4 X-ray bursts The standard model of an X-ray burst is based on a close binary system, where one member of this system is a neutron star and the other one a hydrogen-rich star. Neutrons in a neutron star are so densely packed that they form a degenerate gas. Because of the high density and the deep gravitational well of a neutron star, a freely falling proton will arrive at the surface of the neutron star with an energy greater than MeV [4]. At the surface, the high-energy proton will lose its energy via emission of many X-rays, mostly with an energy range below 2 kev. At the surface of the neutron star, the accumulated material becomes degenerate like in novae. Under high temperature hydrogen starts to burn via the hot CNO cycle (see section.2). This process is rapid and occurs under a constant pressure up to the moment when degeneracy of the neutron gas is lifted and it expands. When the cooling rate becomes equal to the energy production, X-ray bursters reach a peak-surface temperature of 2.5 T 9. At the end, most of the initial helium and most of the other isotopes are converted into heavy isotopes with mass heavier than A=72, up to Sn, see Ref. [5]. In these processes proton-rich isotopes beyond 56 Fe can be produced..2 CNO cycles The cold CNO cycle is the fusion process of 4 hydrogen nuclei into helium with an energy production of MeV per cycle. This process takes place in hot (above.6 T 9 ), hydrogen-rich environment where small amounts of heavier elements (C, N, O) act as catalysts, and thus their relative abundances remain unchanged during the process. The cold and hot CNO cycles are presented in Fig... The cold CNO cycle operates at temperatures below.2 T 9 and it is governed by the slowest reactions. These are the β + decays of 3 N and 5 O. When the stellar temperature reaches.2 T 9, proton capture on 3 N is more probable than β + decay and the hot CNO cycle becomes operational. When the stellar temperature increases above.4 T 9, an additional hot CNO cycle becomes available via the (α,p) reaction on 4 O. At stellar temperatures beyond.5 T 9 and.8 T 9, the 5 O(α,γ) 9 Ne and 8 Ne(α,p) 2 Na reactions become possible, respectively. These two reactions provide a break out from the CNO cycle into the NeNa cycle. Davids et al. [6] pointed out that there is no significant break out from the CNO cycle via the 5 O(α,γ) 9 Ne reaction. The 8 Ne(α,p) 2 Na reaction proceeds, at the temperatures required for explosive burning of hydrogen in novae and X-ray bursts, through individual resonances above the α- emission threshold in the compound nucleus 22 Mg. Therefore, to calculate the rate for this reaction, one has to know the properties of these high-lying resonances.

12 .3. NeNa and MgAl cycles 5 The Hot CNO Cycle 2 C 5 N Cold CNO Cycle.2 T 9 2 C 3 N 4 O 4 N 5 O.4 T 9 5 N 3 C 7 F 8 Ne 8 F 5 O 4 N 2 Na 9 Ne.8 T 9.5 T 9 (p,γ) β + (p,α) (α,γ) (α,p) Figure.: Cold and hot CNO cycles. Different nuclear reactions are marked with different arrow types..3 NeNa and MgAl cycles In hot stellar environments, where temperatures are higher than those for the stable CNO cycles, other cycles become operational where nuclei with a mass heavier than oxygen start to act as a catalyst. These are the NeNa and MgAl cycles, which form a sequence as presented in Fig..2. These two cycles do not contribute significantly to the stellar energy production, because of the higher Coulomb barriers involved in the reactions presented in Fig..2. But they are important for the production of elements between 2 Ne and 27 Al. 22 Na and 26 Al are two long-lived nuclei with a half-life of 2.62 years and years, respectively. Their decay is followed by γ-ray emission at.275 MeV and.89 MeV, respectively; see Fig..3. Therefore, if these two isotopes can survive nova and X-ray burst explosions, their characteristic γ-ray emissions can be observed after the nova explosion. Weiss and Truran [7] concluded that a nova can be an important source of 26 Al in the Galaxy, and that some nearby novae can produce amounts of 22 Na which can be detected with γ-ray observatories. The detection of the.275 MeV and.89 MeV γ-rays will provide an excellent bench-

13 6. Introduction NeNa cycle MgAl cycle 2 Ne 22 Na 25 Mg 26 Al 2 Na 22 Ne 25 Al 26 Mg 27 Si 2 Ne 23 Na 24 Mg 27 Al (p,γ) β + (p,α) Figure.2: NeNa and MgAl cycles. Different nuclear reactions are marked with different arrow types. 5 + β + t /2 =7.2x 5 yr 3 + t /2 =2.62 yr 22 Na g.s. 26 Al g.s. β E γ =.89 MeV E γ =.275 MeV 26 Mg g.s. 22 Ne g.s. Figure.3: 26 Al g.s. and 22 Na g.s. β + decay followed by γ-ray emission. mark for existing novae models. The.89 MeV γ-ray, associated with the β + decay of 26 Al nucleus, has been observed by the COMPTEL γ-ray observatory based on board of the CGRO satellite [8]. The very same observatory could not detect any signal of the 22 Na.275 MeV γ-ray [8]. However, the upper limit for the ejected amount of 22 Na from observed novae could be determined.

14 .4. Astrophysical importance of 22 Na and 26 Al 7.4 Astrophysical importance of 22 Na and 26 Al Na The cold NeNa cycle is presented in the left panel of Fig..2. From this figure we can see that the production of 22 Na follows the 2 Na(β + ν) 2 Ne(p,γ) 22 Na reaction path. The proton capture reaction on 2 Na becomes more probable than its β + - decay with rising stellar temperatures. Consequently, the 2 Na(p,γ) 22 Mg(β + ν) 22 Na reaction path may lead to the production of 22 Na. Furthermore, proton capture can occur on 22 Mg (t /2 =3.87 s) at these temperatures. Due to the low Q-value for the 22 Mg(p,γ) 23 Al reaction, the majority of 23 Al nuclei will, however, be destroyed via the inverse 23 Al(γ,p) 22 Mg reaction [9]. The lack of relevant data for the 2 Na(p,γ) 22 Mg reaction is one of the main sources of uncertainty in calculations of the production of 22 Na. Therefore, the study of the nuclear structure of 22 Mg has been subject of many recent experiments [,, 2, 3, 4, 5, 6]. These studies together with the results from capture reactions using radioactive beams of 2 Na [7, 8, 9] have contributed to a better understanding of the production rate for 22 Na in stellar environments. In Section.2 we mentioned that the 8 Ne(α,p) 2 Na reaction might provide a possible break-out from the hot CNO cycle to the NeNa cycle. This reaction proceeds at high temperatures, which can be found in explosive stellar environments, through individual resonances above the α-emission threshold in the compound nucleus. Therefore, the 8 Ne(α,p) 2 Na reaction and the nuclear structure of 22 Mg have been the subject of several experimental investigations. Caggiano et al. [3] and Chen et al. [2] measured excitation energies in 22 Mg above the α-emission threshold with an accuracy between 5 kev and 45 kev. Bradfield-Smith et al. [2] and Groombridge et al. [2] measured resonance strengths and the excitation energies of levels in 22 Mg above MeV with an accuracy of 5 kev and 4 kev, respectively. However, the accuracy achieved in these 4 experiments for the determination of the excitation energy is not satisfactory for the astrophysical rate calculations. The errors in the excitation energies produce more than a 5% error in the calculated 8 Ne(α,p) 2 Na reaction rates. In the present work we measured excitation energies of levels in 22 Mg with an error less than 5 kev for levels below.5 MeV and approximately 2 kev for levels above.5 MeV Al The.89 MeV γ-ray line was observed by the HEAO-C satellite γ-ray observatory in 984 [22]. The discovery of 26 Al in the interstellar medium [23] demonstrated that indeed nucleosynthesis processes are ongoing in our present time. The half-life of yr for

15 8. Introduction 26 Al is short compared to the time scale of the galactic chemical evolution. The production 26 Si + + t /2 =6.4 s E x =.229 MeV 25 Al 5/ Al g.s. 2 + t /2 =476 fs E x =.89 MeV 25 Mg 5/ Mg g.s. (p,γ) (β + ) Figure.4: 26 Al g.s. and 26 Al m production and β + decay schemes. Different nuclear reactions are marked with different arrow types. mechanism for 26 Al depends on the stellar environment, the most promising production sites being Wolf-Rayet stars, AGB stars [24], novae, and supernovae explosions. The two 26 Al production mechanisms are presented in Fig..4. The first one can occur at lower stellar temperatures above.35 T 9, where 26 Al is produced via the reaction sequence: 25 Al(β + ν) 25 Mg(p,γ) 26 Al g.s. (β + ν) 26 Mg (γ) 26 Mg g.s.. The half-life of the ground state of 26 Al is yr. Its decay to the first-excited 2 + state in 26 Mg is followed by γ-ray emission of.89 MeV. However, if proton capture on 25 Al is faster than 25 Al β + decay (which will occur at stellar temperatures higher than.4 T 9 ), the reaction will follow the second path; 25 Al(p,γ) 26 Si(β + ν) 26 Al (β + ν) 26 Mg g.s.. In this path 26 Si decays into the isomeric, first-excited + state of 26 Al with a half-life of only 6.4 s, which subsequently decays to 26 Mg g.s.. Following this path there will be no.89 MeV γ-ray emission, and galactic production of 26 Al cannot be observed. Ward and Fowler [25] showed that for temperatures lower than.4 T 9 there is no thermal equilibrium between 26 Al g.s. and the first-excited state and consequently, they can be treated like separate species. The 25 Al(p,γ) 26 Si reaction has not been measured directly, yet. However, the nuclear structure of 26 Si has been the subject of several recent experimental studies [26, 27, 28,

16 .5. Outline of this work 9 29]. In this context we also performed the 28 Si(p,t) 26 Si measurements above the protonemission threshold with an energy resolution of 3 kev. At stellar temperatures above.3 T 9, energy production and nucleosynthesis in an explosive hydrogen-burning environment are determined by the rp-process and αp-process. The proton-capture reaction rates are orders of magnitude faster than β-decay rates in the rp-process. The reaction path consists of a series of (p,γ) reactions up to a nucleus where further proton capture is inhibited. The proton-capture reactions can be inhibited by the negative proton-capture Q-value, which is followed by proton decay, or by a small positive proton-capture Q-value, which is followed by a photodisintegration process. The reaction flow has to wait for the slow nucleus β-decay and that nucleus is denoted as a waiting point. The 22 Mg, 26 Si, 3 S, and 34 Ar isotopes are β + unstable and possible waiting points. These isotopes are important because most of the reaction flow is passing through them. Because of the small (p,γ) Q-value for 22 Mg, photodisintegration prevents a significant flow through a subsequent 23 Al(p,γ) reaction. However, this waiting point can be bridged by the 22 Mg(α,p) 25 Al reaction. This reaction is part of a chain 4 O(α,p)(p,γ) 8 Ne(α,p)(p,γ) 22 Mg(α,p)(p,γ) 26 Si(α,p)(p,γ) 3 S(α,p)(p,γ) 34 Ar(α,p)(p,γ) 38 Ca and has been theoretically investigated by Fisker et al. [3]. These calculations were performed to explain bolometrically double-peaked type I X-ray bursts, see Refs. [3, 32, 33]. The 22 Mg(α,p) 25 Al reaction has not been experimentally investigated previously, because there are no 26 Si data above the α-emission threshold which is located at 9.64 MeV. We performed the 28 Si(p,t) 26 Si measurements and for the first time observed the nuclear structure above the α-emission threshold in 26 Si..5 Outline of this work In Chapter 2 the formalisms for the calculation of the nuclear reaction rates are given. The experimental set up used for our (p,t) reaction studies is explained in Chapter 3. The momentum calibration for the obtained spectra is presented in Chapter 4, whereas the analysis of the data obtained for 22 Mg and 26 Si are explained in Chapters 5 and 6, respectively. In Chapter 7, the conclusions and an outlook of this work are given.

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18 Chapter 2 Theoretical model of stellar reaction rates In this chapter we will discuss the basic theoretical model necessary to calculate stellar reaction rates. The complete formalism and the notation are taken from Ref. [4]. In the following natural units will be used. Here, we will discuss only reaction rates induced by a charged particle. 2. Stellar reaction rates The non-degenerate stars have a simple structure. The hottest and most dense region is the core. Going from the interior outwards, the stellar temperature and pressure drop. A normal non-degenerate star consists of a plasma containing nuclei and free electrons. The plasma in the stellar core is fully ionized, and consists of various isotopes. Because this plasma is in thermodynamic equilibrium, the velocity distributions of the nuclei can be described by Maxwell-Boltzmann velocity distributions, see Ref. [4], φ(υ) = 4πυ 2 ( m 2πkT ) ) 3/2 exp ( mυ2 2kT This equation can be written in terms of the kinetic energy of the nucleus: ( φ(e) E exp E ) kt (2.) (2.2) where the most probable value of the kinetic energy is equal to kt. The total reaction rate for a nuclear reaction a + B c + D can be written as: R = N a N B συ ( + δ ab ) (2.3) where N a and N B are the numbers of particles of type a and B per cubic centimeter in a stellar plasma, respectively. The factor (+δ ab ) prevents double counting in case identical particles interact with each other. Here, συ = φ(υ)υσ(υ)dυ (2.4)

19 2 2. Theoretical model of stellar reaction rates is referred to as the reaction rate per particle pair, where σ(υ) is the nuclear cross section, and φ(υ)dυ is the probability that the relative velocity υ between the particles involved in the nuclear reaction is between υ and υ+dυ. The integration for exothermic reactions extends from υ= up to infinity, and for endothermic reactions the integration starts at the threshold velocity for a particular reaction. As mentioned before, for nuclear reactions taking place in the stellar interior, the distribution of the relative velocity v of the interacting nuclei a and B is described by the Maxwell-Boltzmann velocity distribution, and the reaction rate per particle pair is given as: συ = φ(υ a )φ(υ B )υ a υ B σ(υ)dυ a dυ B. (2.5) After a proper kinematic transformation, using the reduced mass µ and total mass M, and integration over the center-of-mass velocity (the cross section depends only on the relative velocity between the interacting particles) we obtain the following equation: συ = ( ) /2 8 πµ (kt) 3/2 ( σ(e)e exp E ) de (2.6) kt 2.2 Non-resonant reaction rates (direct reaction rates) A nucleus is a positively charged entity, and therefore, two colliding nuclei repel each other with a force proportional to the product of their respective nuclear charges. This repulsive force leads, in the case of a charged particle undergoing an attractive nuclear interaction, to a potential barrier called the Coulomb barrier. For the fusion of two charged particles they have to penetrate through the Coulomb and centrifugal barriers; see Fig. 2.. The total repulsive potential is given V = Z BZ a e 2 r + l(l + ) 2 2µr 2 (2.7) where Z B and Z a are the nuclear charges of the interacting particles, r is their mutual distance, µ is the reduced mass of the projectile-target system, and l is the orbital angular momentum. The typical Coulomb barrier for the interaction between two light nuclei is of the order of a few hundred kev or higher. At a temperature of.5 T 9, which is typical for a star like the Sun, energies of the nuclei are of the order of kev. However, in case of the highest predicted supernovae temperature of 9 T 9, the corresponding energies for the nuclei are of the order of a few hundred kev. Therefore, the typical particle energies in the stellar environment are smaller than the repulsive Coulomb potential between the nuclei. However, Gamow [34] showed that nuclei can penetrate the barriers with a small but finite probability via the quantum-tunneling effect. The penetrability P l through the Coulomb

20 2.2. Non-resonant reaction rates (direct reaction rates) 3 Figure 2.: Schematic view of the combined nuclear, Coulomb, and centrifugal potentials. Where R C(E) is a classical turning point for the Coulomb and centrifugal barriers. Figure taken from Ref. [4]. and centrifugal barriers is energy dependent and can be expressed as: P l (E, R n ) = Fl 2(E, R n) + G 2 l (E, R n) (2.8) where F l and G l are, respectively, the regular and the irregular solutions for the Coulomb wave function for a given relative angular momentum, at the nuclear interaction radius of: R n =.35 (A /3 a + A /3 ) fm (2.9) where A a and A B are the masses of the projectile and target, respectively, given in atomic mass units. Non-resonant reactions are reactions with a one-step process, where a direct transition into a bound state occurs. Radiative capture, presented in Fig. 2.2, is one example of a non-resonant reaction. Other possible non-resonant reactions are: pickup and stripping reactions, Coulomb excitation, and charge-exchange processes. B

21 4 2. Theoretical model of stellar reaction rates Radiative capture B(a,γ)D E cm a+b γ Q Figure 2.2: Radiative-capture reaction as an example for a non-resonant reaction. D In case of non-resonant reactions, the cross section is proportional to a single matrix element. In our example for radiative capture it is given as: σ γ D H γ B + a 2 (2.) where H γ is an electromagnetic operator describing the transition. At a relative kinetic energy much smaller than the Coulomb barrier and for an orbital angular momentum l=, the tunneling probability can be approximated as [4]: P = e 2πη (2.) The quantity η is called the Sommerfeld parameter and it is equal to η = Z BZ a e 2 ħυ (2.2) The interaction cross section is dependent on the penetrability and the de Broglie wave length, which describes the geometrical effects of the cross section σ πλ 2 πk 2. Including all these contributions, we can write the cross section as: σ(e) = E S(E)e 2πη (2.3) The factor e 2πη describes the penetration through the Coulomb barrier of point-like nuclei

22 2.2. Non-resonant reaction rates (direct reaction rates) 5 CROSS SECTION (log. scale) S(E) FACTOR (lin. scale) Figure 2.3: An example of the cross section and the astrophysical S-factor for a charged-particle non-resonant nuclear reaction. Figure taken from Ref. [4]. without orbital angular momentum (s-waves). The function S(E) is called the astrophysical S-factor and contains all the nuclear physics of the reaction. In non-resonant reactions, the cross section varies continuously as a function of the energy, see Fig For astrophysical applications we are usually interested in a cross section at an incident energy of a few kev where data usually do not exist. The astrophysical S- factor function S(E) varies smoothly as a function of the energy for non-resonant reactions in the region well below the Coulomb barrier, see lower panel in Fig 2.3. Because of this characteristic feature, the astrophysical S-factor is used to extrapolate measured cross sections to energies relevant for the astrophysical environment. By combining Eqs. 2.3 and 2.6 we obtain the equation for the reaction rate for non-resonant stellar nuclear reactions συ = ( ) /2 8 πµ (kt) 3/2 ( S(E) exp E kt b ) de (2.4) E /2 where the quantity b arises from barrier penetrability and is given as: b = (2µ) /2 πe 2 Z B Z a /. (2.5) Since for non-resonant reactions, the astrophysical S-factor varies slowly, the strongest influence on the reaction rate is caused by the exponential penetrability term ( b ) and E /2

23 6 2. Theoretical model of stellar reaction rates Figure 2.4: The Gamow peak is indicated with the shaded area. Figure taken from Ref. [4]. the exponential Maxwell-Boltzmann term ( E kt ). The exponential factor which is related to the penetrability through the Coulomb barrier shifts the effective distribution of the reaction rates to a higher energy E. The convolution of these two exponential functions results in a peak of the integrand near the energy E, which is usually much larger than kt, known as the Gamow peak; see Fig For a given stellar temperature T, nuclear reactions can take place in a relatively narrow region of energies around E. Because S(E) varies slowly as a function of the energy, it can be approximated by a constant value over the Gamow peak: and the reaction rate has the form: συ = S(E) = S(E ) = constant, (2.6) ( ) /2 8 πµ (kt) S(E ) 3/2 ( exp E kt b ) de. (2.7) E /2 By taking the first derivative of this formula, the position of the Gamow peak for some

24 2.3. Resonance reaction rates through narrow resonances 7 Resonant radiative capture B(a, γ )D E R E r Γ Q a+b γ D Figure 2.5: Resonant radiative capture as an example of resonant reactions. Q=m a+m B m D is the reaction Q-value. E R is the center-of-mass projectile energy needed to populate the centroid of a resonance state. D nuclear reaction can be determined. It is given by the following formula: E = ( bkt 2 The width of the Gamow peak is given by: ) 2/3 =.22(Z 2 B Z2 a µ)/3 T 2/3 9 MeV (2.8) E = 4 3 E kt =.237(Z 2 BZ 2 aµ) /6 T 5/6 9 MeV (2.9) In Section 2.5 we will show the relevant parameters for the Gamow windows for the 8 Ne(α,p) 2 Na, 2 Na(p,γ) 22 Mg, 25 Al(p,γ) 26 Si, and 22 Mg(α,p) 25 Al reactions discussed in this work. 2.3 Resonance reaction rates through narrow resonances A resonant process is a two-step process, in which an excited state E r of the compound nucleus is formed that subsequently decays into lower-lying states. The resonant reaction shows a rapid variation of the cross section over a small energy range. An example of a resonant reaction is presented in Fig The reaction cross section for resonant reactions is proportional to two matrix elements: σ γ D H γ E r 2 E r H D B + a 2. (2.2) Where the matrix element involving the operator H D describes the formation of the compound state E r, and the second matrix element describes the subsequent γ-decay. This process can happen only if the energy of the entrance channel matches closely with the

25 8 2. Theoretical model of stellar reaction rates energy of the resonance involved: E R + Q = E r. (2.2) This process can occur for all excited states above the threshold energy Q, when E R satisfies the condition given by Eq In addition, a resonant state can be formed via a given reaction channel if selection rules are fulfilled (angular momentum and parity conservation laws). Resonance phenomena often occur in physical systems. The cross section for a resonant reaction can be written in the form σ(e) Γ a Γ c (E E R ) 2 + (Γ/2) 2 (2.22) using the analogy with a damped oscillator driven by an external force. Γ a and Γ c are the partial widths of the entrance and exit channels, respectively, and Γ is the total width of the resonant state. The cross section is exactly given by the Breit-Wigner formula: σ BW (E) = πλ 2 2J + (2J a + )(2J B + ) ( + δ Γ a Γ c ab) (E E R ) 2 + (Γ/2) 2 (2.23) Where λ is the de Broglie wave length and J, J a and J B are the spins of the resonant state, of the projectile, and of the target, respectively. The term ω = 2J + (2J a + )(2J B + ) ( + δ ab) (2.24) is known as the spin statistical factor, which can be obtained by summing over all final states and averaging over initial states. The summing over the final states reflects that the probability for a given process increases with an increasing number of available final states. Because in the entrance channel the colliding nuclei can have (2J a +) and (2J B +) substates, the factor (2J a+)(2j B+) reflects the probability that in the entrance channel the nuclei are in one particular substate. The criterion for narrow resonances is that the resonance width is much smaller than the resonance energy: Γ E R. Ref. [4] presents a quantitative criterion: Γ E R %. (2.25) An example of a narrow resonance is given in Fig Under this circumstance the Maxwell-Boltzmann function changes very little over the resonance region and the term E exp ( ) E kt in Eq. 2.6 can be approximated by ER exp ( ) ER kt and taken outside the integral in Eq. 2.6:

26 2.3. Resonance reaction rates through narrow resonances 9 Figure 2.6: Schematic view of the Maxwell-Boltzmann energy distribution at a given stellar temperature T together with the cross section for a narrow resonance. Figure taken from Ref. [4]. συ = ( ) /2 ( 8 πµ (kt) E 3/2 R exp E ) R σ BW (E)dE (2.26) kt For the integration of the Breit-Wigner cross-section yield for a narrow resonance, we can neglect the energy dependence of λ, Γ, Γ a and Γ c : where the product σ BW (E)dE = π 2 λ 2 2J + (2J a + )(2J B + ) ( + δ ab) Γ aγ c Γ ωγ = ω Γ aγ c Γ = 2J + (2J a + )(2J B + ) ( + δ ab) Γ aγ c Γ (2.27) (2.28) is referred to as the strength of the resonance. Low-energy narrow resonances well below the Coulomb barrier will have a larger probability for decay by γ-rays as compared to particle decay. For example, in case of the (p,γ) reaction we have Γ γ Γ p and Γ γ Γ. Consequently the resonance strength depends on the partial decay width by proton emission ωγ = ω ΓγΓp Γ ωγ p. However, in case the resonance energy is well above the Coulomb barrier, Γ p will be much larger than Γ γ, and Γ γ will be the dominant factor in the resonance strength.

27 2 2. Theoretical model of stellar reaction rates The particle partial width can be calculated as shown in Ref. [9] as Γ particle = 3 2 µrn 2 P l C 2 S particle (2.29) where µ is the reduced mass of the interacting particle, S particle is its spectroscopic factor (in our case proton or alpha), C is the isospin Clebsch-Gordan coefficient, and P l is the penetrability through the Coulomb barrier and centrifugal barrier for orbital-momentum l evaluated at an interaction radius R n ; see Eq To calculate the particle penetrability through the Coulomb and centrifugal barriers we used the code PENE [35]. In the case where several narrow resonances contribute to the reaction rates, their contributions are simply summed to obtain the reaction rates for a particular reaction; συ = ( ) 3/2 2π ħ ( 2 (ωγ) i exp E ) i µkt kt i (2.3) We obtained this formula by combining Eqs. 2.26, 2.27 and Inserting all the relevant quantities, the reaction rate can be calculated by the formula: N A συ =.54 5 (µt 9 ) 3/2 i (ωγ) i exp(.65e i /T 9 ) [cm 3 s mol ] (2.3) where the resonance strength (ωγ) i is in units of ev, and the resonance energy E i is in units of MeV. 2.4 Reactions through broad resonances Here, we consider resonances which are broader than the relevant energy window for a given stellar temperature. According to the definition given in Ref. [4], a broad resonance is a resonance where Γ E R %. (2.32) The cross section σ(e) extends over a large range of energies, and the dependence of the cross section on energy needs to be taken into account for the calculation of the stellar reaction rate according to Eq The energy dependence of the cross section can be calculated as: σ(e) = σ R E R E Γ a (E) Γ c (E) (Γ R /2) 2 Γ a (E R ) Γ c (E R ) (E E R ) 2 + Γ 2 (E)/4 (2.33) where the cross section and the total width are known at the resonance energy: σ R = σ(e R ), Γ R = Γ(E R ). Obviously, knowledge of the energy dependence of the partial

28 2.4. Reactions through broad resonances 2 widths is necessary for the calculation of the cross section. In general, charged particles need to penetrate Coulomb and centrifugal barriers. The particle partial width can be calculated by Γ l (E) = 2 R n ( ) /2 2E P l (E, R n )θl 2, (2.34) µ where the quantity θ 2 l is called the reduced width of the nuclear state, which represents the probability of finding the excited state in the configuration l. Usually θ l is determined experimentally. The penetrability factor is given in Eq With an increasing orbital angular momentum l, the centrifugal barrier becomes larger than the Coulomb barrier and Γ l drops rapidly. As an example, the calculated partial proton widths for different values of l for the 6 O+p 7 F reaction are presented in Fig (ev) (MeV) Figure 2.7: The calculated partial proton width Γ l for the reaction channel 6 O+p 7 F as function of proton energy for values of the orbital angular momentum l= to 6. Figure taken from Ref. [4]. The energy dependence of Γ γ is given as: Γ γ (E γ ) = α L E 2L+ γ (2.35) where L is the multipolarity of the emitted γ-ray and α L is constant for each multipolarity

29 22 2. Theoretical model of stellar reaction rates depending on the nuclear structure of the resonance. Still, the energy dependence of the partial γ-widths is not as strong as for particle emission, because of the particle penetrability through the Coulomb barrier. Partial γ-widths are of the order of ev or smaller. In contrast, the particle widths can be very small well below the Coulomb barrier, and they can be in the order of MeV above the Coulomb barrier. If the resonance is near or above the Coulomb barrier, the partial particle width varies little over the resonance region (E=E R ± Γ R /2); see Fig In contrast, for a resonance well below the Coulomb barrier, the partial particle width for the entrance channel varies very rapidly. On the other hand, the partial particle width for the outgoing channel, in case of (α,p) reactions discussed here, varies more slowly, because the emitted particle has an energy, which is increased by the amount of the Q-value of the reaction. The cross section is not symmetric with respect to E R because it varies much more for energies below the resonance energy E R as compared to energies above the resonance energy. 2.5 The astrophysically relevant excitation-energy regions for 22 Mg and 26 Si In the previous sections we discussed direct reactions and the simplest two resonant reactions, which are important for the calculation of the stellar reaction rates. The formalism given in the previous sections will be enough to calculate rates for the 8 Ne(α,p) 2 Na, 2 Na(p,γ) 22 Mg, 25 Al(p,γ) 26 Si, and the 22 Mg(α,p) 25 Al reactions discussed in this thesis. The Gamow window concept is directly applicable in the case of direct reactions. However, it is useful for resonant reactions to calculate the position of the Gamow window for a particular stellar temperature T and to see in which excitation-energy region a particular resonance will dominate. In Section.2 we already mentioned that at a stellar temperature above.8 T 9 the 8 Ne(α,p) 2 Na reaction becomes possible. This reaction is one of the possibilities to link reactions between the hot CNO cycle and the NeNa cycle. This breakout from the hot- CNO cycle gives the energy trigger for the X-ray bursts. For the more accurate X-ray bursts models it is therefore necessary to obtain more precise data for 22 Mg resonances for the entire span of stellar temperatures up to 2.5 T 9. On the right side of Fig. 2.8 we present the positions of the Gamow windows (peak position and width) for the 8 Ne(α,p) 2 Na reaction at.8 T 9 and 2.5 T 9. In Section.4. we discussed the astrophysical importance of the 2 Na(p,γ) 22 Mg reaction as a tool to check present novae models. Since the novae peak temperatures reach values of.4 T 9, we are interested in 22 Mg levels up to 5.95 MeV. For the X-ray bursts we are interested in temperatures up to 2.5 T 9. On the left side of Fig. 2.8 we show the Gamow window for this reaction for the temperature of 2.5 T 9. From Fig. 2.8 it can be seen that we are interested in the 22 Mg nuclear structure from the proton-emission threshold up to.5

30 2.5. The astrophysically relevant excitation-energy regions for 22 Mg and 26 Si 23 MeV. In Section.4.2 we already mentioned that 25 Al(β + ν) 25 Mg(p,γ) 26 Al g.s. can occur at a stellar temperature above.35 T 9 and 25 Al(p,γ) 26 Si(β + ν) 26 Al at higher stellar temperatures. Because supernovae explosions can be possible sources for 26 Al production we are interested in the 25 Al(p,γ) 26 Si reaction rates for temperatures up to the supernovae peak temperature of 4 T 9. On the left side in Fig. 2.9 we indicate Gamow windows at 2.5 T 9 and 4 T 9 for the 25 Al(p,γ) 26 Si reaction at the X-ray bursts and supernovae peak temperatures, respectively. From the same figure it can be seen that for this reaction we are interested in 26 Si levels up to 8 MeV. The 22 Mg(α,p) 25 Al reaction was not previously studied. On the left side of Fig. 2.9 Gamow windows for this reaction are shown for temperatures of T 9, for the X-ray bursts peak temperature 2.5 T 9 and supernovae peak temperature 4 T 9. From the same figure it can be seen that for this reaction we are interested in 26 Si levels up to 3 MeV.

31 24 2. Theoretical model of stellar reaction rates 2 Na+p 8 Ne+α MeV.272 MeV 2.5 T 9.8 T MeV 8 Ne + α 2.5 T MeV 2 Na + p 22 Mg Figure 2.8: Relevant astrophysical windows for the 8 Ne(α,p) 2 Na (right) and 2 Na(p,γ) 22 Mg (left) reactions at temperatures of.8 T 9 and 2.5 T 9 for the 8 Ne(α,p) 2 Na reaction and at a temperature of 2.5 T 9 for the 2 Na(p,γ) 22 Mg reaction. The full horizontal lines indicate the thresholds for proton and α emission, respectively. The dashed lines indicate for illustrative purposes the positions of two known levels in 22 Mg.

32 2.5. The astrophysically relevant excitation-energy regions for 22 Mg and 26 Si Mg + α 25 Al + p 3 MeV 4 T T MeV 22 Mg + α T 9 4 T T MeV 25 Al + p 26 Si Figure 2.9: Relevant astrophysical windows for the 22 Mg(α,p) 25 Al (right) and 25 Al(p,γ) 26 Si (left) reactions at temperatures of T 9, 2.5 T 9 and 4 T 9 for the 22 Mg(α,p) 25 Mg reaction and at temperatures of 2.5 T 9 and 4 T 9 for the 25 Al(p,γ) 26 Si reaction. The full horizontal lines indicate the thresholds for proton and α emission, respectively. The line at 3 MeV excitation energy is marked for illustrative purposes.

33

34 Chapter 3 Experimental setup and method The astrophysics motivation to study the nuclear structure of 22 Mg and 26 Si is given in Chapter 2. The (p,t) reaction was used for the following reasons: The Q-values of the (p,t) reaction on 24 Mg and 28 Si are strongly negative in comparison to those on other stable target nuclei, namely 2.97 MeV and 22.9 MeV, respectively. Only 2 C and 6 O have similar Q-values. The outgoing triton has a very high mass-to-charge ratio ( m =3) resulting q in a magnetic rigidity large compared to other reaction products. Therefore, depending on the proton bombarding energy, mainly tritons pass through the magnetic spectrometer. In this experiment, in addition to the lines resulting from the (p,t) reaction only four other lines were observed resulting from 3 C(p,d) and 7 O(p,d) reactions. 3. Experimental area The experiments 24 Mg(p,t) 22 Mg and 28 Si(p,t) 26 Si were performed at the Research Center for Nuclear Physics (RCNP) of Osaka University using a MeV proton beam from the Ring Cyclotron. The proton beam impinged on a.82 mg/cm 2 thick 24 Mg target and a.7 mg/cm 2 thick 28 Si target. Both targets were self-supporting but contained 2 C and 6 O impurities. For the identification and subtraction of events from these contaminants, a mg/cm 2 thick 2 C target and a mg/cm 2 thick Mylar target were used. Although these impurities were sources of unwanted background they provided important information for the energy calibration. The Grand Raiden (GR) spectrometer was used for the momentum analysis of the outgoing tritons, with its standard detector system consisting of two 2-dimensional Multi-Wire Drift Chambers (MWDC) followed by three plastic scintillators for timing and particle-identification purposes. The MWDC detectors allowed for the determination of the horizontal and vertical positions of the tritons and consequently their angles of incidence at the focal plane (see for further details Section 3.5). The layout of the experimental facility at RCNP is depicted in Figure 3.. A primary beam is extracted from the AVF Cyclotron and injected into the Ring Cyclotron (RC) for further acceleration. The protons of MeV were transported through the West-South (WS) beam line leading to the target chamber of the GR spectrometer located in the West experimental hall.

35 28 3. Experimental setup and method Figure 3.: Floor plan of the experimental hall at RCNP. Figure is taken from Ref. [36]. The new fully dispersion-matched WS beam line of the RCNP facility was designed for high-resolution spectroscopy experiments. In our case the resolution was mainly limited by the target thickness rather than the resolving power of the spectrometer; see Wakasa et al. [37]. 3.2 The GR spectrometer The GR spectrometer has been designed for high-resolution spectrometry measurements at RCNP. A detailed description of the GR spectrometer can be found in Ref. [38]. A schematic view of the layout of the GR spectrometer is shown in Fig It consists of two dipole magnets (D and D2), two quadrupole magnets (Q and Q2), one sextupole magnet (SX) and one multipole magnet (MP). The function of the sextupole magnet is to minimize

36 3.2. The GR spectrometer 29 Figure 3.2: Schematic view of the spectrometer. Figure taken from Ref. [39]. the second-order ion-optical aberrations, (x θ 2 ) and (x φ 2 ) and to keep the horizontal focalplane tilt angle at 45. The multipole magnet can generate magnetic fields with quadrupole, sextupole, octupole and decapole components. Its function is to compensate for other ionoptical aberrations of the system. For spin-physics studies the last DSR magnet is added for the measurement of the in-plane polarization transfer. This dipole was not excited in the present experiment. The spectrometer can momentum analyze particles with a maximum rigidity of Bρ max =5.4 Tm and has a remarkably high momentum resolving power of p/ p = D x M x x = 37 (3.) for a monochromatic beam spot with the size of x = mm (Ref. [37]), where M x is the magnification and D x the dispersion of the spectrometer. The design parameters of the spectrometer are listed in Table 3.. Thus, with the momentum dispersion and the

37 3 3. Experimental setup and method Table 3.: Design parameters for the GR spectrometer taken from Refs. [37, 38]. Mean orbit radius 3 m Total deflection angle 62 Angular range 4 to 9 Momentum dispersion (D x=(x δ)) 5.45 m Momentum resolution p/ p 37 Momentum range 5% Tilting angle of focal plane 45 Focal-plane length 2 mm Maximum magnetic rigidity 5.4 Tm Maximum field strength (D,D2).8 T Horizontal magnification (M x=(x x)).47 Vertical magnification (M y=(y y)) 5.98 Horizontal acceptance angle ±2 mrad Vertical acceptance angle ±7 mrad Maximum solid angle 4.3 msr Flight path of central ray 2 m horizontal magnification listed in Table 3., the momentum resolving power for x = mm is p/ p=37. The observed spectral resolution with the old (WN) beam line was significantly less, mainly due to the momentum spread of the incident beam. Because the GR spectrometer is characterized by a large momentum dispersion D x =5.45 m and a magnification M x =.47, a very large momentum dispersion of D tgt = D x /M x 37 m at the target position is required in order to achieve lateral dispersion-matching. The new WS beam line is designed to achieve this beam-line dispersion D tgt. Consequently, the beam spot has a large horizontal size leading to uncertainties in the determination of the scattering angle. These uncertainties are coming from different scattering angles, which depend on the position on the target. Therefore, angular dispersion-matching is required to be able to reconstruct from measured focal-plane parameters the scattering angle with good angular resolution. The new WS beam line is designed to fullfill all required matching conditions: focus condition, lateral dispersion-matching, and also angular dispersion-matching. Details of the WS beam line is given in Ref. [37]. 3.3 Matching between beam line and spectrometer In this section we will use transport notation following papers by Fujita et al. [4] and Wakasa et al. [37]. A charged particle is transported from the exit of the cyclotron to the target and, after scattering, through the spectrometer up to the focal plane where the detector system is positioned. Particles at the exit of the RC have the following horizontal coordinates x, θ and δ, denoting the position, angle and momentum deviations from the central trajectory, respectively. A particle ray with these coordinates is transported

38 3.3. Matching between beam line and spectrometer 3 by the beam line, represented by the transport matrix B, to the target location. There the coordinates are given by x, θ and δ. At the target, the coordinates are transformed by the T, defined by T, K, C, Eqs. 3.4, 3.5 and 3.6 below to x 2, θ 2 and δ 2, and through the spectrometer by the spectrometer matrix S to the coordinates x fp, θ fp and δ fp in the focal plane. The x fp and θ fp coordinates of a particle at the focal plane can be described in first order with the B and S matrix elements and T (Ref. [37]): x fp = (s b T + s 2 b 2 )x +(s b 2 T + s 2 b 22 )θ +(s b 6 T + s 2 b 26 + s 6 C)δ +(s 2 + s 6 K)Θ +higher-order terms (3.2) θ fp = (s 2 b T + s 22 b 2 )x +(s 2 b 2 T + s 22 b 22 )θ +(s 2 b 6 T + s 22 b 26 + s 26 C)δ +(s 22 + s 26 K)Θ +higher-order terms (3.3) Where the suffixes, 2 and 6 refer to the x, θ and δ parameters, respectively. The parameter T is called the target function T = cos(α Φ T )/cos(φ T ). (3.4) The parameter α is the angle between the central ray of the incident beam at the target position and the central ray in the spectrometer and Φ T is the angle between the normal direction to the target and the direction of the incident beam (see Fig. 3.3). The factor Θ = θ θ 2 has the meaning of an effective scattering angle, and indicates how much the scattering angle of the particle is different from the nominal scattering angle α (see Fig. 3.3), where θ is the angle of an incident proton particle relative to the central ray of the beam, and θ 2 is the angle of the outgoing triton particle relative to α. The factor K is the first-order kinematic factor given by: The parameter C is the dispersion-matching factor K = (/p out )( p out / α) (3.5) C = (p in /p out )( p out / p in ) (3.6)

39 32 3. Experimental setup and method where p out is the momentum of the outgoing particle at the target position, and p in is the incoming momentum of the particle at the target position. For elastic scattering C=. Particle ray 2 Target θ 2 X 2 Central ray into spectrometer Particle ray θ β α X. Φ T Central ray of beam Figure 3.3: Schematic representation of the scattering of one particle with coordinates (x, θ ) relative to the central ray of the beam. The (x 2, θ 2) are coordinates of outgoing particles and α is the scattering angle of the central ray. The true scattering angle is defined by: β = α + (θ 2 θ ) (3.7) The minimum size of the image x fp in the focal plane and thus the best resolution is achieved when the coefficients of x, θ and δ in Eq. 3.2 are zero. For a good definition of the scattering angle θ fp, we require that the coefficient δ in Eq. 3.3 should be zero. The process of adjusting the spectrometer to eliminate the Θ term in Eq. 3.2 (K= s 2 s 6 ) is called the kinematic correction. Proper kinematic correction is achieved when particles with different θ fp converge at the same location x fp. By requiring in addition that the coefficient of δ in Eq. 3.2 and Eq. 3.3 is zero we obtain the following conditions for the lateral dispersion and for the angular dispersion b 6 = s 6 s ( + s s 26 K s 2 s 6 K) C T (3.8) b 26 = (s 2 s 6 s s 26 )K, (3.9) at the target location, needed to optimize the momentum resolution and the angular resolu-

40 3.3. Matching between beam line and spectrometer 33 Figure 3.4: Schematic view of ion trajectories under different matching conditions (a) achromatic beam transportation, no matching; (b) lateral dispersion-matching Eq. 3.8, ambiguity of the angular information still remains; (c) lateral and angular dispersion-matching Eqs. 3.8 and 3.9 are achieved. Figure is taken from Ref. [4]. tion at the focal plane of the GR spectrometer. From Eqs. 3.8 and 3.9 it can be seen that the parameters b 6 and b 26 of the beam line should depend on the parameters s 2 and s 6 needed for the kinematic correction of the spectrometer. The resolving power of the matched system is given by: R = (/2x )(s 6 /M ov ) (3.) where M ov = (s b T + s 2 b 2 ) has the meaning of an overall magnification and if the matching conditions are satisfied it has the form: M ov = (s b T s 6 b 2 K) (3.) The matching conditions are schematically illustrated in Fig An achromatic beam impinging on the target is illustrated in Fig. 3.4 (a). A particle with a momentum difference p from momentum of the central ray is dispersed by the spectrometer and the resolution is limited by the beam momentum spread. Under lateral dispersion-matching, shown in Fig. 3.4 (b), a particle with a momentum difference p from momentum of the central ray hits the target at a different position in such way that the beam s dispersion is compensated by the spectrometer s dispersion. But a particle with a momentum difference p from that of the central ray is crossing the focal plane under different angles θ fp, so that an angular measurement in the focal plane does not allow the determination of the scattering angle at the target. When both lateral and angular dispersion-matching conditions have been achieved simultaneously, Fig. 3.4(c), the position and the angle of the particle measured at the focal plane (x fp, θ fp ) do not depend on the beam dispersion p and the measured angle in the focal plane corresponds to a unique scattering angle within measurement limitations.

41 34 3. Experimental setup and method 3.4 Over-focus mode In order to determine the angular distribution we need to perform an angular measurement with good resolution. Near the scattering angle depends equally on both the vertical and horizontal angle components. Consequently, we need to achieve a good resolution in both directions. The GR spectrometer has a small vertical-angle magnification of.7. Because of this small magnification, an ejectile with a large vertical scattering angle at the target will have a small angle at the focal plane. Under these conditions it is impossible to achieve a vertical resolution at the target better than 2 mrad. The seemingly simple method of achieving a better angle measurement by increasing the distance between the first and the second position measurement, does not lead to much improvement due to multiple scattering in the first detector and the resulting poor position measurement in the second detector. Therefore, in order to improve the vertical-angle resolution, the ion-optical properties of the GR spectrometer in the vertical direction were considered in the transfer matrix formalism: y fp = (y y)y 2 + (y φ)φ 2 + (y yx)y 2 x 2 +(y yθ)y 2 θ 2 + (y yδ)y 2 δ +(y φx)φ 2 x 2 + (y φθ)φ 2 θ 2 +(y φδ)φ 2 δ + higher-order terms (3.2) where y 2 and φ 2 are the vertical position and the outgoing angle, respectively, of a particle at the target (similarly for x 2 and θ 2 in the horizontal direction), and (y y),(y φ) etc. are matrix elements of the spectrometer matrix, that transforms coordinates from the target to the focal-plane coordinates [39]. In Fig. 3.5 the vertical trajectories are depicted for three different settings: (a) Focus mode of GR: the term (y φ) from Eq. 3.2 is equal to zero for the central ray (ρ=3 cm). (b) Over-focus mode: the strength of the Q magnet is increased and (y φ) >. (c) Under-focus mode: the strength of the Q magnet is decreased and (y φ) <. Because of the finite (y φ)φ 2 term in Eq. 3.2 in the over-focus and under-focus modes, particles with different outgoing angles φ 2 at the target are transported to different positions y fp in the focal plane. Therefore, it is possible to determine φ 2 values from the measured values of y fp. In order to calibrate the over-focus mode, the (y φ) and (y φθ) terms should be known. All other terms in Eq. 3.2 are expected to make the relationship between y fp and φ 2 more ambiguous, and for a useful angle determination they need to be small. The largest ambiguity comes from the vertical beam spot size at the target because of a large vertical magnification (y y) 6. The horizontal beam spot size also induces an ambiguity.

42 3.4. Over-focus mode 35 Figure 3.5: The vertical trajectories of the scattered particle with the vertical position y 2 = ± mm and the outgoing angle φ 2 = ±.46 mrad from the target: (a) in the normal point-to-point focus mode of GR, (b) in the over-focus mode, (c) in the under-focus mode. Figure is taken from Ref [39]. The accurate determination of φ 2 from y fp measurements depends on how much we can increase the value of the (y φ) term by a proper adjustment of the strength of the Q magnet, and how small the ambiguities can be made in practice. A detailed description of the new ion-optical mode called off-focus mode applied to the GR spectrometer at RCNP can be found in Ref. [39]. This experiment was performed in the over-focus mode. For the reconstruction of the coordinates at the target position from the measured coordinates at the focal plane, calibration measurements with a multi-hole aperture (sieve-slit) were performed.

43 36 3. Experimental setup and method MWDC MWDC2 st PLASTIC ( mm) PL nd 2 PLASTIC (3 mm) PL2 rd 3 PLASTIC ( mm) PL3 Al ( mm) Al2 ( mm) Figure 3.6: Schematic view of the focal-plane detector system of GR spectrometer. 3.5 The detection and data-acquisition systems For our (p,t) experiment we used the standard focal-plane detector system which exists at RCNP. This system consists of two 2-dimensional multi-wire drift chambers (MWDC s) located at the focal plane for the determination of the positions and the angles of the particles. These MWDC s are followed by a set of three plastic scintillation detectors. In between these plastic scintillator detectors we placed aluminum plates to prevent secondary electrons produced by one scintillator to hit another scintillator. Furthermore, the thickness of the first two plastic scintillators and the aluminum plates were chosen in such a way that the highest energy tritons produced in the (p,t) reaction are stopped before they reach the third plastic scintillator detector. The third scintillator detector was therefore used as a veto detector to eliminate the background from scattered protons with a larger range. The layout of the focal-plane detector system is depicted in Fig The signals from the first two scintillators were used for triggering of the system as well as particle identification through energy-loss measurements in these detectors. The specifications of the MWDC s are given in Table 3.2. Each MWDC has two anodewire planes (X and U) sandwiched between three cathode planes. The anode planes consist of sense wires and potential wires. Fig. 3.7 shows the schematic structure of the X-wire plane. Sense wires are mounted at distances of 6 mm. For the U-wire plane the pitch of the wire spacing is 4 mm. The purpose of the potential wires is to produce a more uniform

44 3.5. The detection and data-acquisition systems 37 Table 3.2: The specifications of the MWDC. Wire configuration X( vertical), U(48.2 ) Active area 5 mm 2 mm Number of sense wires 92 (X), 28 (U) Cathode-anode gap mm Anode-wire spacing 2 mm Sense-wire spacing 6 mm (X), 4 mm (U) Sense-wire thickness 2 µm gold-plated tungsten wire Potential-wire thickness 5 µm gold-plated beryllium-copper wire Cathode µm carbon-aramid film Gas mixture Argon + Iso-butane + Iso-propyl-alcohol (7.4%), (28.6%) Entrance and exit window 2.5 µm aramid film Distance between two MWDC s 25 mm Pre-amplifier LeCroy 2735DC electric field between the anode plane and cathode plane. The electrons created during the ionization of the detector gas by charged particles can only produce an avalanche near the sense wires. The drift time of the electrons is measured in order to determine a chargedparticle trajectory. In the example shown in Fig. 3.7, the particle trajectory is determined by using the drift-time information from four wires. For the proper determination of the trajectory it is necessary to have information from a minimum of three wires. By combining the information from the MWDC s, it is possible to determine a trajectory more accurately. The gas used in the MWDC is a mixture of 7.4% argon and 26.6% iso-butane, and a small amount of iso-propyl-alcohol. The iso-propyl-alcohol is added to reduce deterioration of the detector gas, which can lead, e.g., to polymerization of the gas on the wire surfaces. As signal preamplifier and discriminator LeCroy 2735DC cards are directly connected (without any cables) to the MWDC s. The ECL output signals from these 2735DC cards were transferred to LeCroy 3377 TDC s, which digitize the signals of each wire. The scintillator signals were digitized by a LeCroy FERA and FERET system (see Fig. 3.8). After readout, the events were stored in high speed memory buffer modules (HMS s) in the VME crate, and copied to a VMIC5576 reflective memory module (Rm5576 in VME crate) by an MC684 based CPU board with the OS/9 operating system. The data from the experimental room were transported via a fiber-optic link to another Rm5576 module in the counting room, and analyzed with a SUN work station. Then, the data are transferred from the SUN work station to the IBM RS/6 SP station via an FDDI line. Finally, the data are stored in list mode on a large hard disk.

45 38 3. Experimental setup and method Figure 3.7: Schematic structure of an X-plane of the MWDC. A typical track of a charged particle is illustrated. Figure is taken from Ref. [4]. Event Build Recording On line analysis Figure 3.8: Schematic view of the data-acquisition system used in the (p,t) experiment. Figure is taken from Ref. [42]. 3.6 Experimental settings for the (p,t) reaction As mentioned in the introduction, the goal of this experiment was to obtain high-resolution spectra at and also at more backward angles. Because of these requirements we decided to perform a measurement at using an existing Faraday cup in the first dipole D of the

46 3.6. Experimental settings for the (p,t) reaction 39 Target Figure 3.9: Schematic view of the first dipole magnet of Grand Raiden with existing Faraday cups; figure by Berg [43]. GR spectrometer and at 8 and 7. For the latter two angles we used a Faraday cup inside the target chamber. With the required experimental conditions at, none of the existing Faraday cups at their standard locations inside the dipole magnet was able to collect the proton beam. However, the existing ( 3 He,t) Faraday cup (see Fig. 3.9) could be used in our experiment by slightly moving it downstream. Nevertheless, we still had a huge background at, caused by the proton beam. This background could be reduced significantly by changing the angle of the GR spectrometer to.3, still covering the angles around. In order to ensure the correct measurement of the integrated charge of the beam in this Faraday cup, it was checked against measurements in the other beam-stop inside the target chamber. In our data, we still had a triton background from reactions on the Faraday cup, but we were able to remove it by using the information from the time of flight of the particles. Using this setup, the spectra in the veto scintillator showed that we did not have any protons that were able to pass through the GR spectrometer. Therefore, we did not include the veto scintillator in the trigger logic. We used beam intensities of 2 na and 5 na, for the measurements at.3 and the

47 4 3. Experimental setup and method backward angles, respectively. The smaller current at.3 is related to the (p,t) reaction when the Faraday cup inside of the D magnet of the GR spectrometer was used, which introduced a high counting rate in the detectors. The live time of the system varied between 82% and 98% during this experiment. Table 3.3: Angular and magnetic-field settings of the GR spectrometer. Angle [ ] B [T] B2 [T] B3 [T] Three magnetic-field settings were used to cover the whole excitation-energy region of interest with partly overlapping spectra. Table 3.3 lists the angular and the magneticfield settings. Table 3.4 shows the excitation-energy regions of interest for the 24 Mg and 28 Si targets covered by the different magnetic fields. A spectrum will be indicated by the target ( 24 Mg, 28 Si, carbon or Mylar), the GR angle (.3, 8 or 7 ) and the magneticfield settings (B, B2, B3). As an example, the overlapping spectra for the 24 Mg(p,t) 22 Mg reaction at a scattering angle of 8 are shown in Fig. 3.. Table 3.4: The excitation-energy range for the different magnetic-field settings for 24 Mg and 28 Si targets, respectively target B B2 B3 24 Mg -.5 up to 6.5 MeV 5.5 up to 2. MeV 7.5 up to 4. MeV 28 Si -.3 up to 5.7 MeV 4.7 up to.2 MeV 6.8 up to 3.2 MeV

48 3.6. Experimental settings for the (p,t) reaction 4 24 Mg(p,t) 22 Mg excitation energy at GR angle 8 o 24 Mg(p,t) 22 Mg Θ=8 ο B excitattion energy [MeV] Excitation energy (MeV) B2 excitation energy [MeV] Excitation energy (MeV) B3 Excitation energy (MeV) excitation energy [MeV] Counts/2 counts/2 kev kev Counts/2 counts/2 kev kev Counts/2 counts/2 kev kev Figure 3.: 24 Mg(p,t) 22 Mg excitation-energy spectra obtained at three magnetic-field settings and a scattering angle of 8. Overlapping areas can be seen in this figure.

49

50 Chapter 4 Calibration As explained in Chapter, the 22 Mg and 26 Si levels above the proton-emission threshold (5.542 MeV and 5.52 MeV, respectively) up to an excitation energy of about 3 MeV are of astrophysical interest. We performed measurements covering the excitation-energy region from the ground state up to 3 MeV, and used the low-lying well-known levels of 22 Mg for momentum calibration of the spectra. The reaction 24 Mg(p,t) 22 Mg was chosen for the momentum calibration because the uncertainty of the mass of 22 Mg (.27 kev) is smaller than for 26 Si ( kev), and there are more accurate energy levels available for 22 Mg. 4. Software corrections Owing to the kinematics of the reactions, particles emerging from the target at the same excitation energy of the residual nucleus but at different reaction angles have different momenta. This, together with the optical aberrations, can influence the resolution if the angular acceptance is large enough. The plots on the left side of Fig. 4. show the effects of kinematics and aberrations on the energy spectra of our experiment. In order to achieve the best possible resolution for our (p,t) spectra we performed software corrections to compensate for these effects. Preceding this, sieve-slit measurements have been made to correlate target coordinates with focal-plane coordinates; see Ref. [44]. In the analysis we used a procedure outlined by Yosoi [45]. The correction parameters are determined by using distributions of the horizontal projection, θ fp, and vertical projection, φ fp, of the scattering angle, Θ scatt, measured in the focal plane versus the position, x fp, in the focal plane: θ fp versus x fp and φ fp versus x fp. The left panel of Fig. 4.(a) shows the 2-dimensional spectrum θ fp versus x fp. Every curved vertical line corresponds to a different discrete state. In order to remove these curvatures a fourth-order polynomial function was employed. The right panel of Fig. 4.(a) shows the same spectrum after corrections and straight vertical lines can be observed. Similarly the left panel of Fig. 4.(b) shows correlations of φ fp versus x fp. In order to correct these effects a second-order polynomial function was employed. The right panel of Fig. 4.(b) shows the same spectrum after corrections. The effect of these corrections on the resolution for our (p,t) spectra is illustrated in Fig. 4.(c) where we show the 24 Mg(p,t) 22 Mg position spectra before and after perform-

51 44 4. Calibration ing the corrections for the.3 scattering angle. Because the kinematic effects are larger for the larger scattering angles, the effects of these corrections are also much larger at these backward angles. In case of full angular acceptance (.2 < θf p <.2 and 3.2 < φf p <3.2 ), we fitted the ground state of 22 Mg for the uncorrected and the corrected spectra obtained at.3 scattering angle with a Gaussian function. The obtained resolutions (FWHM) were 43 kev and 9 kev for the uncorrected and the corrected spectra, respectively. From the difference in the achieved resolution, it can be seen that the software corrections employed for the kinematics and the aberrations are crucial to achieve high resolution spectra. However, by using these corrections we were not able to achieve fully straight θf p versus xf p and φf p versus xf p spectral lines, for the whole angular acceptance along the used region used region Counts/ mm φfp (deg) θ fp (deg) Uncorrected Corrected (a) x position (mm) x position (mm) (b) x position (mm) x position (mm) (c) x position (mm) x position (mm) Figure 4.: (a) and (b): Plots of the horizontal scattering angle θf p and the vertical scattering angle φf p versus the position in the focal plane before and after corrections. (c) The 24 Mg(p,t)22 Mg position spectrum before and after corrections measured with the spectrometer at.3.

52 4.2. Background subtraction 45 Time of flight (channels) background x position at focal plane (mm) Figure 4.2: The two-dimensional time-of-flight versus x-position spectrum for the measurement at.3 and the B2 magnetic-field setting. The background due to tritons from the Faraday cup is removed by setting a gate as shown. The characteristic level structure for the 24 Mg(p,t)22 Mg reaction is visible in the lower part of this 2-dimensional display. entire focal plane. It can be seen from Fig. 4. that still substantial aberrations occur for.6 > θf p and θf p >.4 and φf p >.5. Therefore, we made a restriction on the data in the angular region of.6 < θf p <.4 and.5 < φf p <.5, indicated in the right panels of Figs. 4.(a) and 4.(b), respectively. This improved the energy resolution (FWHM) of the ground state to 3 kev. 4.2 Background subtraction In Section 3.6 we mentioned that at the spectrometer angle of.3 we had a huge background originating from the (p,t) reaction on the Faraday cup inside the GR spectrometer. In the two-dimensional spectrum of the time-of-flight versus xf p it is easy to identify which part is the background (see Fig. 4.2), and by a simple cut on the time-of-flight spectra we removed this background. As mentioned before we used self-supporting targets of 24 Mg and 28 Si with a thickness of.82 mg/cm2 and.7 mg/cm2, respectively. Both targets had impurities of 2 C and 6 O, which have larger (p,t) cross sections and similar reaction Q-values compared to 24 Mg

53 46 4. Calibration 24Mg(p,t) 22 Mg B Θ=.3 ο Counts/.25 mm O g.s. 4 Cg.s O C Position at focal plane (mm) Figure 4.3: The 22 Mg spectrum taken at.3 and the magnetic-field setting B; only strongly populated impurity lines are indicated with their known excitation energies. and 28 Si. Therefore, our spectra are affected by impurity lines from the 2 C(p,t) C and 6 O(p,t) 4 O reactions (see Fig. 4.3). t Counts/ 2.5 kev Counts/ 2.5 kev 24 Mg(p,t) 22 Mg 4 O B2 Θ=.3 ο Excitation energy (MeV) Figure 4.4: Part of the 22 Mg spectrum taken at the magnetic-field setting B2 before and after subtraction of the 4 O contaminant line. Due to evaporation of the impurities during the experiment, the ratio between the 22 Mg lines and contaminant lines was changing, as can be seen by comparing the 6.59 MeV 4 O level in Figs. 4.3 and 4.4 taken at different times.

54 4.3. Reference data 47 By taking additional data on a Mylar and a natural carbon target, we could identify and subtract impurity lines from the 22 Mg and 26 Si spectra. The first step in the subtraction method was to subtract the C lines from the Mylar spectra after normalization using isolated peaks. In this way, we obtained a pure 4 O spectrum without C peaks or continuum. Subsequently, we subtracted the normalized C and 4 O impurity spectra from the 22 Mg and 26 Si spectra. To determine the normalization coefficients for the subtraction method we used the well-separated and the strongly populated impurity levels at low excitation energy (the C g.s. and 4 O g.s. in Fig. 4.3). Fig. 4.4 shows the part of the 24 Mg(p,t) 22 Mg spectrum with E x MeV before and after the subtraction of the 4 O impurity line at 6.59 MeV. In case of the spectra obtained at the B3 magnetic-field settings, which cover the higher excitation-energy regions, lines originating from the 3 C(p,d) 2 C and the 7 O(p,d) 6 O reactions are used to perform background subtraction. The (p,d) lines were easy to remove from our spectra by using the E/E particle identification in the first and second plastic scintillator detectors PL and PL2. Because of the highest statistics and the smallest influence of kinematics, the excitation energies were determined from the measurements at.3. In case where an impurity line is covering a level in the triton spectrum taken at.3, the excitation energy for that level was determined from the spectra at 8 or Reference data The first magnetic-field setting (B) covers the energy range in 22 Mg approximately from.5 MeV up to 6.8 MeV. Fig. 4.5 represents the 24 Mg(p,t) 22 Mg spectrum obtained with B=.7846 T at an angle of.3. Because of its high statistics, this spectrum was chosen for the momentum calibration. For the calibration of Bρ/Bρ central only 24 Mg(p,t) 22 Mg lines were used. Their excitation energies were taken from Ref. [6]. We used the most accurate energy levels below the proton-emission threshold that were published by Seweryniak et al. [6], who studied the high resolution 2 Na(p,γ) 22 Mg reaction. We used six strongly populated calibration lines (see Fig. 4.5) indicated by their excitation energies. These lines are distributed approximately from 46 mm up to 36 mm along the focal plane, which is active from 6 mm to 6 mm in total. The levels from Ref. [6] used for the Bρ/Bρ central calibration are indicated with the superscript in the second column in Table 4.. The third column in this table shows the data by Bateman et al. []. The levels indicated with the superscript a are from Endt [46], and were used by Bateman et al. [] for their calibration. Differences exist between the following values of Refs. [6] and []: (4.42(3) (42), 5.7() (2)), although the first two values agree within the error bars. It is important to

55 48 4. Calibration mention that Bateman et al. [] used only two previously known 22 Mg levels at 5.37 MeV and MeV for the calibration of their (p,t) spectra. But the MeV level is in discrepancy with the corresponding level listed in Ref. [6] and the 5.37 MeV level has a three times larger error than the corresponding level listed in Ref. [6]. In order to correct Bateman s energy calibration a simple linear function was employed. The corrected energies are shown in the fourth column of Table 4.. The new calibration gives a better agreement between the excitation energies of Bateman et al. [] and Seweryniak et al. [6]. These energies will be compared with our results rather than those from Ref. []. Mg(p,t) Mg E p =98.7 MeV g.s. B ο Θ=.3 C g.s. 4 Og.s. Peak Peak position position [mm](mm) Count/.25 mm Figure 4.5: The 24 Mg(p,t) 22 Mg spectrum obtained with magnetic-field setting B at an angle of.3. Note: Lines used in the calibration are marked with their excitation energies in MeV. The excitation energies given in this figure are rounded off to three digits beyond the comma

56 4.4. Fits of the spectra 49 Table 4.: Energy levels of 22 Mg from the literature used in the present analysis. All energies are given in MeV. Data from Seweryniak et al. [6] are compared with those from Bateman et al. [], and from Caggiano et al. [3] J π Seweryniak Bateman Bateman Caggiano et al. [6] et al. [] et al. [] et al. [3] 2 Na(p,γ) 22 Mg 24 Mg(p,t) 22 Mg corrected 25 Mg( 3 He, 6 He) 22 Mg + g.s. - - g.s. a (3) a (6) a (3) (42) 4.43(42) 4.49 a (5) 5.37(4) a 5.362(4) 5.33(7) ( + ) 5.893(8) 5.897(7) 5.887(7) 5.94(6) (4) (6) (6) 5.3(4) (4) (4) (6) 5.459(6) 545(5) () 5.739(2) a 5.76(2) a (25) (25) (3) 6.4(3) 6.5(4) (5) 6.24(5) 6.246(4) (3) (5) 6.24(5) 6.246(4) (6) 6.37(6) 6.329(6) (7) 6.66(7) 6.66(4) (4) 6.78(4) 6.77(5) The levels indicated with were used for the present calibration; Levels indicated with a are used by Bateman et al. and Caggiano et al. for their calibrations. 4.4 Fits of the spectra For the determination of the position of the peaks we used two Gaussian functions and a square-hyperbola function as defined in Eq. 4.. This combination gives a lower reduced χ 2 compared to, for example, a pure Gaussian or Breit-Wigner shape. The parameters were obtained by fitting the first-excited state of 22 Mg (.247 MeV). For the region E x < MeV this shape was fixed, i.e. the only free parameters were the peak positions and the heights. This function was used for all states below the proton-emission threshold of MeV, since the instrumental width of the peaks was larger than the natural width of the corresponding levels. Fig. 4.6(a) shows the first-excited state of 22 Mg fitted by our function 4. together with the difference between the data and the fitted function.

57 5 4. Calibration Two Gaussian functions and square hyperbola: f(x) = y exp( (x x)2 ln2) x < x x 2 exp( (x x)2 x 2 2 A x x + ln2) x < x x + η x 2 B (x x ) 2 x > x + η x 2 (4.) with: x Abscissa value of the maximum of the Gaussian function, y Function value at x, x Half width at half maximum of the Gaussian function for x < x, x 2 Half width at half maximum of the Gaussian function for x > x, A, B Parameters to insure that the model function is differentiable along the abscissa, η Distance between the starting points of the square hyperbola and x in units of x 2 (absolute distance = η x 2 ). For levels above the proton-emission threshold the fitting function was changed to a Gaussian and two exponential functions (Eq. 4.2). Gaussian function and two exponential functions: exp( (x x)2 x ln2) x 2 η x < x < x + η 2 x f(x) = y exp(a (B + x)) x x η x exp(a 2 (B 2 x)) x x + η 2 x (4.2) with: x Abscissa of the maximum of the Gaussian function, y Function value at x, x Half width at half maximum of the Gaussian function, A,2, B,2 Parameters to insure that the model function is differentiable along the abscissa, η, η 2 Distances between the starting points of the exponential functions and x in units of x (absolute distance = η i x i, i can be or 2). The shape of the first-excited level above the proton-emission threshold was used to determine η and η 2. This change was introduced to accommodate the natural width that increases above the proton threshold. This particular function allows to fix the tails (η, η 2 ) while the width ( x) can be adjusted. Fig. 4.6(b) shows the first-excited state above the proton-emission threshold of 22 Mg fitted with our function 4.2. For the magnetic-field setting B2 the function 4.2 was fitted to the MeV level in order to determine η and η 2. This line was positioned close to the center of the focal plane and had good statistics (see Fig. 4.7). The same line shape is also used for the spectrum obtained at the magnetic

58 4.4. Fits of the spectra 5 (a) Two Gaussina functions plus a square hyperbola.247 MeV Position (mm/4) Counts Counts Position (mm/4) (b) Gaussian function plus two exponential functions Counts 5.7 MeV Position (mm/4) Counts Position (mm/4) Figure 4.6: Fit for 22 Mg at Θ lab =.3 (a) level at E x=.247 MeV by the function given in Eq. 4. and (b) level at E x= 5.7 MeV by the function given in Eq. 4.2, together with residues of the fits. See Fig field B3; since none of the peaks in this spectrum provides acceptable statistics to determine the parameters η and η 2 listed in Eq After removing the background contributions caused by the (p,t) reaction on the Faraday cup and by contaminants in the target, as explained above, no additional background was assumed to be present in the excitation-energy region below MeV. For higher energies a constant background was assumed. In the 22 Mg spectra, we defined four regions of interest (see Table 4.2) motivated by

59 52 4. Calibration Counts/2.5 kev B Excitation energy (MeV) Θ=.3 24 Mg(p,t) 22 Mg ο Figure 4.7: The 24 Mg(p,t) 22 Mg spectrum at the magnetic field B2 and at.3. Note that the lines used in the calibration are indicated by their excitation energies. See Fig astrophysical reasons and due to the calibration procedure. Because of the low statistics and the increase in the natural width at higher excitation energies, we changed the binning for these regions from about.75 kev/channel in region one to about 4 kev/channel in region four as shown in Table 4.2. Table 4.2: Regions of interest and binning in the spectra. Name Energy Binning calibration g.s MeV.25 mm 2 above of p-emission threshold MeV.5 mm 3 above of α-emission threshold MeV mm 4 above of.5 MeV.5-3. MeV 2 mm

60 4.5. Calibration function Calibration function A correction for the energy loss of tritons in the target is taken into account in the calibration procedure. Our assumption is that on average the 24 Mg(p,t) 22 Mg reaction takes place at the center of the 24 Mg target. Following this approximation, the triton energy loss is calculated for half of the target thickness. The data for this correction are obtained from the code LISE++ written by Tarasov et al. [47]. The recent direct 2 Na(p,γ) 22 Mg measurements performed with a radioactive 2 Na beam by the TRIUMF-ISAC group (Bishop et al. [7]) obtained.257(5) MeV for the resonant energy of the first-excited level above the proton-emission threshold. On basis of the adopted excitation energy 5.74 MeV at that time, Bishop et al. [7] calculated a new value for the mass excess of 22 Mg. However, the most recent 2 Na(p,γ) 22 Mg measurements [6] show that the excitation energy of the first-excited level above the protonemission threshold is 5.7() MeV. Thus, the mass excess for 22 Mg as given by Bishop et al. [7] has been underestimated. Two new experiments have been performed by Mukherjee et al. [48] and Parikh et al. [49] to measure the mass of 22 Mg. The measurements done with the ISOLTRAP Penning-trap mass spectrometer at CERN [48] yield the most accurate value for the 22 Mg mass excess of (27) kev. Because our statistical error is larger than.5 kev for the relevant peak positions, we are not sensitive enough to use the error of.27 kev for the absolute Bρ calibration. We deliberately increased the error up to a value of.55 kev, so we achieved a χ 2 reduced for the calibration function. This.55 kev is used later in our analysis of the 22 Mg mass error. Two types of errors are included in the fitting procedure used to obtain the calibration function: (a) one regarding the peak position and (b) the other related to the absolute Bρ values. The error of the beam energy is not included, neither in the fitting procedure, nor in the calculation of the energy levels. In fact, a variation in the beam energy of ± kev introduces less than. kev difference in the resulting excitation energies. Four impurity lines were used to determine the beam energy, namely the ground states of C, 2 C and 6 O populated through the 2 C(p,t) C, 3 C(p,d) 2 C (at 8 and 7 ) and 7 O(p,d) 6 O reactions, respectively. These impurity lines were found at the magnetic-field setting B3, which covers the higher excitation-energy region. They were extracted by using a E/E particle identification in the PL and PL2 detectors. Panel (a) of Fig. 4.8 shows the spectrum obtained with the Mylar target at 8 and magnetic-field setting B3. Panel (b) shows the (p,d) spectra extracted using the above-mentioned particle-identification cut. For the calibration of Bρ/Bρ central as a function of the triton position in the focal plane x fp we used a quadratic polynomial function. The six strongly populated 22 Mg levels indicated by their excitation energy in Fig. 4.5 were used for the calibration. The code LSH2 [5] was used to obtain the parameters of the calibration function. The

61 54 4. Calibration Position in focal plane (mm) Position in focal plane (mm) Counts/mm Counts/mm 3 C(p,d) O(p,d) 7 (p,d) spectrum (a) (b) Mylar spectrum collected at B3 and o Θ=8 Figure 4.8: Panel (a) shows the Mylar spectrum without the PID for the deuterons. Panel (b) shows the (p,d) spectrum obtained from the Mylar spectrum by using a PID cut for the deuterons. Table 4.3: Differences between experimentally determined and calculated absolute kinetic energies for outgoing deuterons and tritons at E p=98.7 MeV. Reaction Magnetic-field Scattering angle KINEMA Experiment Difference ( ) (MeV) (MeV) (kev) 2 C(p,t) C B C(p,d) 2 C B C(p,d) 2 C B O(p,d) 6 O B calibration function for the magnetic-field setting B and Θ=.3 is: Bρ/Bρ central = (A + B x fp + C x 2 fp )/( T m) (4.3)

62 4.5. Calibration function 55 The parameter x fp is the position in mm along the focal plane. The parameter T m is Bρ central for the B magnetic-field setting. A=( ±.5) 3 T m B=( ±.5) 2 T m/mm C=( 2.8 ±.5) 6 T m/mm 2 The obtained reduced χ 2 in this procedure is.97. Note: The 22 Mg mass uncertainty is deliberately increased up to a value of.55 kev. The proton beam energy of 98.7 MeV was determined such that the discrepancies between the experimentally determined kinetic energies for the outgoing deuterons and tritons (the sixth column in Table 4.3) and the energies calculated with the code KINEMA [5] (the fifth column in Table 4.3) for the same beam energy are minimized. These discrepancies might be a consequence of kinematics (different peak shape) and energy loss caused by unknown deposition of carbon and oxygen in the target. The levels measured at both magnetic-field settings B and B2 and at.3 spectrometer angle are listed in Table 4.4. In the first column, the excitation energies determined at the magnetic-field setting B are listed. These levels were used for the calibration of the magnetic-field setting B2. We used the same Bρ/Bρ central function which was employed for the.3, B setting (Eq. 4.3). For other magnetic-field settings the position of all levels are shifted by a small value dx, to lie on one calibration line. The value dx is obtained by the function: Bρ/Bρ [central]b = (A + B (x fp dx) + C (x fp dx) 2 )/( T m) (4.4) where the coefficients A, B and C are the same as in Eq. 4.3, x fp is the position of the level obtained by fitting the spectra, Bρ [central]b is the magnetic field B. The only unknown value is the small shift dx. The same procedure was used for the calibration of all other spectra (B, B2, and B3 for 8 and 7 ). The resulting energies, for these three levels, in the magnetic-field setting B2 are listed in the second column of Table 4.4. This shows that we do have a consistent parameterization for the momentum calibration over the whole focal plane. Adopted values for the last two levels are shown in the third column of Table 4.4. The averaging process is performed only for those levels for which the excitation energy is measured in the extrapolation area of the present calibration. Furthermore, they are used for the calibration at magnetic-field setting B2 and at magnetic-spectrometer angle.3. This is performed in order to decrease a possible error in the extrapolation of the present calibration. In the error of the excitation energy calculated from Eq. 4.3 only the error for the x position in the focal plane enters as statistical error. The systematic error for the excitation

63 56 4. Calibration energy includes errors originating from the reaction-angle determination and from the mass of 22 Mg (.55 kev), which are added quadratically. The systematic and statistical errors are linearly added to obtain the total error, which is quoted for the present 22 Mg data. Table 4.4: The excitation energies obtained for three levels at the B and B2 magnetic-field settings and their weighted averaged energies. Level energy Level energy Level B setting B2 setting averaged energy 5.7(3) 5.785() 5.7() (4) () (9) 6.352(4) 6.366() 6.36(9) Level energy is given in MeV. this is the calibration value taken from Ref. [6]. For the calibration of the B3 magnetic-field setting we used only two levels with the largest statistics; one at 9.752(24) MeV and the other at.277(5) MeV (see Fig. 4.9). Tables 4.5 and 4.6 list our results up to 7.283() MeV together with reference data. Here, we will discuss the main features of the calibration. The detailed discussion of the observed 22 Mg levels will be presented in the next chapter. Table 4.5: Excitation energies in 22 Mg below 5.7 MeV (the proton-emission threshold is at MeV). J π Present Seweryniak Bateman Caggiano McDonald Chen work et al. [6] corrected et al. [3] et al. [52] et al. [2] + g.s. g.s. - g.s. a g.s. g.s. a (3) a.244(32).2463 a (6) a 3.269(5) a (3) 4.43(42) 4.49 a 4.378(35) 4.48(2) a (5) 5.362(4) 5.33(7) 5.32(3) 5.29(2) a + (5.92(5)) 5.893(8) 5.887(7) - 5.3(35) (23) 5.293(4) (6) 5.3(4) 5.286(3) 5.272(9) (4) (4) (4) 5.459(6) 5.45(5) 5.433(25) () 5.76(2) a 5.699(2) 5.7(3) a Level energy is given in MeV. Values from Ref. [6] used for calibration in the present work. a Used for calibration in the previous articles. In Tables 4.5 and 4.6 the present data are compared with those of Seweryniak et al. [6], Bateman s corrected data, Caggiano et al. [3], and McDonald et al. [52]. A good agreement can be observed between our experimental results and the those of Ref. [6]. However, there remain differences between the present data and previous ones for the levels

64 4.5. Calibration function Mg(p,t) 22 Mg B3 Mg(p,t) Mg Θ=8 Ex=98.7 o MeV o Θ=8 energy [MeV] Excitation energy (MeV) Counts/ counts/ kev kev Figure 4.9: The 24 Mg(p,t) 22 Mg spectrum measured at the magnetic field B3 and a scattering angle of 8. The calibration lines are marked by their excitation energies in MeV. See Fig at 6.43(3) MeV, and 6.24(5) MeV (see Bateman corrected data), above the protonemission threshold. The confidence in our calibration is based on the agreement with the values from Refs. [6] and [52]. Values from Ref. [52] are in agreement with those from Ref. [6]. The calibrations of all other references are based on those of Ref. [46] and as we already showed for Ref. [], these authors used a different energy for the last level in their calibration which introduced a discrepancy with Ref. [6]. The quality of our calibration is further substantiated by the agreement for the (2) MeV level with all previous data [, 2, 3] and the level at 7.283() MeV with the values from Refs. [4, 46, 52]. Moreover, the same calibration is used for our 28 Si(p,t) 26 Si data and our measured data for 26 Si are in agreement with published ones.

65 58 4. Calibration Table 4.6: Energy levels in 22 Mg above the proton-emission threshold ( MeV). J π Present Seweryniak Bateman Caggiano McDonald Chen Berg work et al. [6] corrected et al. [3] et al. [52] et al. [2] et al. [4] () 5.76(2) (2) (8) (25) (2) (8) (3) 6.5(4) - 6.4() 6.59(9) 6.226() (5) a 6.246(4) a (9) a (6 + ) (3) 6.24(5) a 6.246(4) a 6.263(2) 6.255() 6.244(9) a (9) (6) 6.329(6) (7) (2) (9) (7) 6.66(4) () 6.66(9) (2) (4) 6.77(5) 6.77(2) 6.767(2) 6.766(2) (2) (9) () (9) (7) (7) (9) () (6) 7.2(2) 7.69() 7.26(9) Level energy is given in MeV. Values from Ref. [6] used for calibration in the present work. a Unresolved doublet. 4.6 Calibration of 26 Si spectra For the 28 Si(p,t) 26 Si spectra we used the same calibration as for the 24 Mg(p,t) 22 Mg spectra; see Eq Because of the smaller uncertainty in the mass of 22 Mg as compared to 26 Si, the calibration with 22 Mg is more accurate, and resulted in smaller errors. In Table 4.7 we list the reference data for 26 Si, that were used in previous and the present calibration of 28 Si(p,t) 26 Si spectra. We used five strongly populated calibration lines (see Fig. 4.) indicated by their excitation energies. The calibration was performed by shifting the calibration points to the calibration line, using the same procedure as employed with Eq This procedure also provided an excellent test for the calibration that we performed for the 24 Mg(p,t) 22 Mg spectra. For our analysis we decided to use the calibration without the levels at 5.45 MeV and 5.55 MeV. These values are obtained by Bardayan et al. [27], Caggiano et al. [28] and Parpottas et al. [29] by using the doublet at 4.86 MeV as the last calibration point. In the present work we resolved the above-mentioned doublet (see discusion in Section 6.2). Therefore, we decided not to use the levels at 5.45 MeV and 5.55 MeV in our calibration procedure in an attempt to examine their excitation energies without influence of the above-mentioned doublet on the calibration. In this way, we obtain an independent measurement for these levels. From the averaged data determined from the measurements made at.3 and magnetic-field settings B and

66 4.6. Calibration of 26 Si spectra 59 Table 4.7: J π and excitation energies (in MeV) of 26 Si levels relevant for calibration. J π Present work Parpottas Caggiano Bardayan Endt et al. [29] et al. [28] et al. [27] [46] + g.s. g.s. a g.s. a g.s. a g.s (2).7959 a.7959 a.7959 a.7959(2) (4) a a a (4) (3) a a (3) - (3.749(4) a a 3.756(2) (2) (4.93(3)) (28) 4.38(4) 4.44(8) 4.55(2) 4.38() (4) 4.83(4) 4.2(6) 4.55(2) 4.83() (3) a a a 4.446(3) (25) 4.86 a 4.86 a 4.85 a 4.86(2) (25) (4) 5.45(4) 5.4() 5.45(2) (2) (6) 5.29(4) (3) 5.33(2) 4 + 5,56(25) 5.55(4) 5.526(8) 5.55(5) 5.562(28) Used in the present calibration. a Used for calibration in the previous articles. B2 we obtain 5.45(4) MeV and 5.56(2) MeV which are in good agreement with the tabulated data 5.45(4) MeV and 5.55(4) of Ref. [29], respectively. For a more detailed discussion of the observed 26 Si levels see Section 6.2. The calibration of 28 Si(p,t) 26 Si spectra is performed by using the calibration of 24 Mg(p,t) 22 Mg spectra as is explained above. Therefore, the calculated 26 Si excitation energies are more sensitive for a possible error in the beam-energy determination. Changing the value of the beam energy by kev resulted in a change in the 26 Si excitation energies by up to.5 kev for states at high excitation energy. In the error of the 26 Si excitation energies calculated from Eq. 4.3 only the error in the x position in the focal plane enters as statistical error. The systematic error includes errors originating from the reaction-angle determination, error of the mass of 26 Si ( kev) and the error resulting from the uncertainty in the beam energy. These three systematic errors are quadratically added to obtain the total systematic error. The systematic and statistical errors are linearly added to obtain the total error, and this error is quoted for the present 26 Si data. For astrophysical reasons it is important to perform more accurate 26 Si mass measurements, especially as this may influence the value of the proton-emission threshold. This can be decisive to tell whether the 5.55(4) level is above or below the proton-emission threshold. However, this level is well below the Gamow window for the 25 Al(p,γ) 26 Si reaction

67 6 4. Calibration 28Si(p,t) 26Si g.s.* Θ=.3 o.796* Cg.s * 3.332* 4.446* counts/ 2.5 kev Θ=8 o θ=7 o Excitation energy (MeV) Figure 4.: The 28 Si(p,t) 26 Si spectra obtained with magnetic-field setting B and spectrometer angles of.3, 8 and 7. The calibration lines are marked with their excitation energy in MeV. See further Fig. 4.5 for more details. at the lower temperatures (above T 9 =.) where the X-ray burst ignites. Consequently, its contribution will not be taken into account. More information regarding this problem is presented in Ref. [49]. These authors measured accurately the mass of 26 Si, showing that the mass of 26 Si is 6 kev heavier than what was tabulated in Ref. [53]. A direct consequence of this work is that the value of the 26 Si proton-emission threshold decreases from 5.58 MeV to MeV.

68 Chapter 5 22 Mg data and their astrophysical implications In this chapter we will discuss the 24 Mg(p,t) 22 Mg data in detail. Our data will be compared with previous results and possible spin assignments will be given. This information will be included in the 8 Ne(α,p) 2 Na and 2 Na(p,γ) 22 Mg reaction-rate calculations and used in a model describing X-ray bursts and ONe (oxygen-neon) novae. 5. The 24 Mg(p,t) 22 Mg angular distributions To obtain angular-distribution parameters for the (p,t) reaction, we performed measurements at three different GR spectrometer angles (.3, 8 and 7 ). Differential cross sections were calculated using the equation where: dσ dω = A t n t D µ Z p [mb/sr] (5.) Ω Q( τ) Z p is the elementary charge of the projectile. A t is the mass of the target [g/mol]. Ω is the effective solid angle [sr]. n t is the number of counts in Ω. D is the detection efficiency (in our experiment above 82%). µ is the thickness of the target [mg/cm 2 ]. Q is the integrated charge [nc]. τ is the dead time. The deduced differential cross sections were compared with distorted-wave Born approximation (DWBA) calculations performed with the code DWUCK4 [54]. A coupledchannels (CC) calculation including inelastic scattering in the entrance and exit channels using the code CHUCK3 [55] did not change appreciably the shape of the calculated angular distributions. The proton optical-potential parameters for the input channel were taken from Ref. [56]. For the outgoing channel the parameterization of Ref. [57] was used which is based on the analysis of the 26 Mg( 3 He,t) 26 Al reaction. All parameters used in the DWBA and CC calculations are given in Appendix A.

69 Mg data and their astrophysical implications We found a large discrepancy between the measured and calculated angular distributions for the first three 22 Mg levels (g.s. +,.247 MeV 2 + and 3.38 MeV 4 + ). The angular distributions of these three levels are presented in Fig. 5. together with the angular distributions calculated with DWUCK4. In addition, we found a discrepancy in the obtained differential cross section for the 2 C(p,t) C reaction compared to those from the previous experiments [58, 59], where the observed angular distributions are less steep. The reason for this discrepancy remains unknown. Because of the discrepancy between the measured and calculated angular distributions, we performed mirror spin-parity assignments on basis of already known data for 22 Na. For the 22 Na levels for which the spin and parity are not known, we made assignments on basis of theoretical calculations given by Ref. [6]. Because the 24 Mg(p,t) 22 Mg reaction proceeds mainly through a direct reaction mechanism it mainly populates states with natural parity in 22 Mg. Therefore, we assumed that the previously observed levels that have not been observed in our experiment are unnatural-parity or high-spin (natural-parity) states calculated dσ/dω (mb/sr) - -2 g.s. + calculated calculated Θ c.m. (deg) Θ c.m. (deg) Θ c.m. (deg) Figure 5.: The measured and calculated angular distributions for the g.s. +,.247 MeV 2 +, and 3.38 MeV 4 + levels of 22 Mg. See further Fig. 4.5 for more details. The solid line is used to guide the eye.

70 Mg and its mirror nucleus 22 Ne Mg and its mirror nucleus 22 Ne It can be seen from Eq. 2.3 that resonance strength data are necessary for the calculation of the reaction rates. Up to now several experiments [7, 9, 8, 2, 2] have been performed with the aim to measure the resonance strengths for the 2 Na(p,γ) 22 Mg and 8 Ne(α,p) 2 Na reactions directly. They succeeded in measuring the resonance strengths for eight resonances above the proton-emission threshold and eight resonances above the alpha-emission threshold. To investigate the rest of the 22 Mg levels and their influence on the 2 Na(p,γ) 22 Mg and 8 Ne(α,p) 2 Na reaction rates we need spin-parity values, and the partial widths Γ p, Γ γ, or Γ α for these resonances. The source for spin-parity assignments and Γ γ and Γ α data can be mirror levels in 22 Ne, which is a stable nucleus. Therefore, its nuclear structure is much better known than 22 Mg. The main data source for excitation energies and spin-parities of levels in 22 Ne is Ref. [6]. Nevertheless, spin-parity values are not known for a number of levels, within the astrophysical region of interest (below 4 MeV). Consequently, additional spin-parity values have been taken from the theoretical calculations listed in Ref. [6]. These values are indicated by the superscript T in our tables and figures. The main obstacle in our spin-parity mirror assignments is the scarcity of definite spinparity assignments in 22 Mg and the lack of it above the + level at 7.28 MeV. Therefore, spin-parity assignments at higher energies become more and more uncertain. α-spectroscopic factors for 22 Mg were taken from Refs. [2, 2]. In addition, we assumed that corresponding 22 Mg and 22 Ne mirror states above the 22 Ne α-emission threshold have the same α-spectroscopic factor. For this purpose, we used the α-spectroscopic factors for 22 Ne levels above the alpha-emission threshold that are listed in Ref. [62]; see also Section 5.6. For the 2 Na(p,γ) 22 Mg reaction-rate calculations we took for 22 Mg the γ-resonancewidths, Γ γ, of corresponding 22 Ne mirror states and corrected these for the differences in γ-ray transition energies; see Section 5.8. In 22 Ne, the Γ γ level widths for all levels below 9.69 MeV are calculated from their life-times, where known, since for these levels only γ-decay is allowed. Here, we explained why additional data for reaction-rate calculations are needed. For this purpose, spin-parity, Γ γ and Γ α data taken from 22 Ne will be used for mirror levels in 22 Mg. In this way, we attempt to minimize the errors introduced by parameters which are not experimentally obtained for 22 Mg nuclei. The procedures for using these parameters in the reaction-rate calculations are described in the following sections.

71 Mg data and their astrophysical implications Table 5.: The calibration region covering the 22 Mg excitation energies below 5.7 MeV (protonemission threshold 5.52 MeV), with the adopted spin-parity assignments. J π Present work Ref. [6] Ref. [] Ref. [3] Ref. [52] Ref. [2] adopted (p,t) (p,γ) (p,t) ( 3 He, 6 He) ( 3 He,n) ( 6 O, 6 He) present + g.s. g.s. - g.s. a g.s. g.s. a g.s (3) a.244(32).2463 a.2478(3) (6) a 3.269(5) a 3.382(6) (3) 4.43(42) 4.49 a 4.378(35) 4.48(2) a 4.42(29) (5) 5.362(4) 5.33(7) 5.32(3) 5.29(2) a 5.346(5) + (5.92(5)) 5.893(8) 5.887(7) - 5.3(35) (8) (23) 5.293(4) (6) 5.3(4) 5.286(3) 5.272(9) () (4) (4) (4) (4) 5.459(6) 5.45(5) 5.433(25) (4) () 5.76(2) a 5.699(2) 5.7(3) a 5.7(5) All energies are in MeV. Levels from Ref. [6] used in our calibration. e Used for calibration in previous articles. Data in the fourth column are the corrected data of Bateman et al., which we already discussed in Section 4.3 Table (4.). 5.3 Calibration region (g.s MeV) In this section we will discuss the unnatural-parity states up to 5.7 MeV. The naturalparity states have already been discussed in Section 4.5 and were used in the energy calibration. In Fig. 5.2 spectra obtained at.3, 8 and 7 are shown with calibration levels indicated by *. It can be seen that in addition to the six calibration levels taken from Ref. [6], we observed three weakly excited levels at 5.92(5) MeV, (23) MeV and 5.454(4) MeV. These three levels correspond to the 5.893(8) MeV +, 5.293(4) MeV 4 + and (4) MeV 3 + states from Ref. [6], respectively. In Table 5. we see an excellent agreement between our data and all previous results. We did not observe the unnatural-parity 2 level at 5.296(4) MeV from Ref. [6]. The only unnatural-parity states which we observed are the 5.92 MeV + and MeV 3 + with very low statistics, as can be seen in Fig In Fig. 5.3 the mirror assignments for levels below the proton-emission threshold are presented. The only astrophysically important state in this interval is the state at 5.7() MeV, which will be discussed in the next section together with the other levels above the protonemission threshold.

72 5.4. Region above the proton-emission threshold (5.542 MeV MeV) Mg(p,t) 22 Mg 4 O g.s. g.s. *.2472* 3.382* Cg.s. 4.42* 5.354* * Θ=.3 ο Counts/ kev Θ=8 ο ρ Θ=7 ο Excitation energy (MeV) Figure 5.2: 24 Mg(p,t) 22 Mg spectra encompassing the calibration region and taken at spectrometer angles.3, 8 and 7. The calibration lines are marked with. The determined excitation energies for 22 Mg are listed in the second column of Table Region above the proton-emission threshold (5.542 MeV MeV) The excellent energy resolution of about 3 kev (FWHM for the ground state) achieved in our experiment allowed us to separate close peaks and to clarify some uncertainties from previous experiments. The presently measured 22 Mg excitation energies between the proton-emission and alpha-emission thresholds are listed in column 3 of Table 5.2. In Fig. 5.4 we show our triton spectra for this energy range taken at spectrometer angles.3, 8, and 7.

73 Mg data and their astrophysical implications present (p,t) (MeV) 5.454(4) (23) (5.92(5)) 5.354* adopted energy (MeV) (4+) * * * g.s.* g.s. + g.s Mg 22 Ne Figure 5.3: The possible 22 Mg mirror assignments for states below the proton-emission threshold. The 22 Mg spin-parity assignments without bracket are taken from previous experiments [6]. The full arrows indicate the mirror assignments between 22 Mg and 22 Ne levels for which spin-parity values are already known. The 22 Ne spin-parity values are taken from Ref. [6]. These spin-parity assignments will be used for reaction-rate calculations. Adopted levels with a dashed line have not been resolved in this experiment but their excitation energies have been taken from the literature. Previous experimental results in this region are listed in columns 4 - of Table 5.2. Our results agree with previously reported excitation energies. In the first column spinparity values from previous experiments are listed as adopted in Ref. [6]. In the second column of Table 5.2 we list the spin-parity values obtained by mirror assignment from 22 Ne, see Fig Note that the superscript T means that the spin-parity assignments for some 22 Ne levels are not known, and are taken from Ref. [6]. In Fig 5.5 we show 22 Mg levels and the corresponding 22 Ne mirror states up to the

74 5.4. Region above the proton-emission threshold (5.542 MeV MeV) 67 alpha-emission threshold. The problem in the spin-parity assignments is the lack of 22 Mg spin-parity data; the highest excitation energy for a state with known spin and parity is that of the + state at MeV. For all levels above this energy mirror assignments are very uncertain. In the following and in later sections we will discuss some of the levels individually based on the following criteria: ) levels which differ significantly in excitation energies from those reported in the literature; 2) levels that are of astrophysical importance; and 3) levels belonging to doublets that have not been resolved earlier (8) MeV (3 ): Our measured excitation energy for this level is lower compared to previous experiments, but still in agreement within errors. Spin-parity of this level remains uncertain. Because this state is strongly populated in the (p,t) experiments we can assume that it has natural-parity. Seweryniak et al. [6] suggest spin-parity 3, which corresponds well with the 5.9 MeV, 3 state in 22 Ne (see Fig. 5.5) () MeV (4 + ): Since Bateman et al. [] measured a peak at 6.24 MeV with a relatively large width of 26±6 kev; they suggested that it consisted of a doublet. This conjecture was confirmed by Seweryniak et al. [6] when they resolved the doublet by measuring a 6 + state at MeV. In the present (p,t) experiment we observed only a level at the lower energy of 6.226() MeV with a width of 3 kev corresponding to the energy resolution of the present experiment. This level probably corresponds to the 22 Ne 4 + state at MeV (Fig. 5.5). The 6 + state at MeV is not observed in the present experiment as expected because of the high momentum transfer necessary to populate this level strongly. 6.36(9) MeV (3 + ) or ( + ): This level is reported in Refs. [3, 8, 9] at an energy above MeV. Our data are consistent with the corrected data of Bateman et al. (5 th column of Table 5.2). In our reaction this peak is weakly excited and in addition affected by the 2 C and 6 O contaminant lines; at the spectrometer angles of.3 and 7 this peak coincides with the contaminant peaks. Bateman et al. [] reported a natural parity for this level. The weak population is an indication of an unnatural-parity state. The possible mirror level in 22 Ne is the MeV 3 + state (Fig 5.5); it has a mirror energy shift which is consistent with the lower lying 3 + state at MeV. However, Fortune et al. [63] and Ruiz et al. [9] correlated this level, on basis of Thomas-Ehrmann shift calculations, to be the mirror of the MeV + state in 22 Ne. Taking into consideration our assignments the 22 Ne MeV state would be the first uncorrelated unnatural-parity 22 Ne state in Fig. 5.5.

75 Mg data and their astrophysical implications Mg(p,t) Mg 5.7* Θ=.3 ο Counts/ 2.5 kev Θ=8 ο Θ=7 ο Excitation energy (MeV) Figure 5.4: The 22 Mg spectra above the proton-emission threshold. A indicates a peak used for calibration. The excitation energy in MeV for each peak is marked in the specific spectrum, where it is determined, obtained at either.3, 8 or 7. All 2 C(p,t) C and 6 O(p,t) 4 O contaminant peaks have been subtracted. The determined excitation energies for 22 Mg are listed in the third column of Table 5.2. See further Fig. 4.5 for more details.

76 5.4. Region above the proton-emission threshold (5.542 MeV MeV) 69 present (p,t) (MeV) 8.7(4) 7.926(5) adopted energy (MeV) 8.62 (3+) T 8.5 (3 ) 7.92 (2+) ( ) (3 ) 8.62 (3+) T (4+) (2+) 7.74(2) (29) 7.389(2) 7.338(3) 7.283() 7.79(8) 7.6(7) 7.45(7) 7.27(9) 6.876(2) (2) 6.62(9) 6.578(7) (4+) (2 ) 7.6 (2+) (3 ) (2+) ( ) 7.6 (3 ) 7.45 (4+) 7.27 (3+)T ( ) (,) (2+) 6.58 ( ) (3+) T (5+) (3,4) (,) (9) 6.226() 6.36(9) (8) (3+) (6+) (4+) 6.37 (3 ) (6+) * (4) (23) (4+) (5) 5.89 (+) * Mg 22 Ne Figure 5.5: The 22 Mg mirror assignments for states between 5 MeV up to the alpha-emission threshold located at 8.4 MeV. The 22 Mg spin-parity assignments without brackets are taken from previous experiments [6]. The 22 Mg spin-parity values within brackets are possible mirror assignments. The full (dashed) arrows indicate definite (tentative) mirror assignments. The 22 Ne spin-parity values are from Ref. [6]. The spin-parity values marked by the superscript T are from Ref. [6]. These spin-parity assignments will be used for reaction-rate calculations. See further Fig. 5.3 for more details.

77 Table 5.2: Excitation energies, spins and parities of levels in the region ( MeV) above the proton-emission threshold. J π J π present Ref. [6] Ref. [] Ref. [3] Ref. [52] Ref. [2] Ref. [4] Ref. [8] Ref. [9] adopted ado. a mirror b (p,t) (p,γ) (p,t) ( 3 He, 6 He) ( 3 He,n) ( 6 O, 6 He) ( 4 He, 6 He) ( 2 Na, 22 Mg) ( 2 Na, 22 Mg) present c () 5.76(2) (2) (5) - 5.7(5) d (8) (25) (2) (5) (6) (8) (3) 6.5(4) - 6.4() 6.59(9) 6.42(3) (8) (4 + ) () (5) g 6.246(4) g (9) g 6.242() g () e (6 + ) (6 + ) (3) 6.24(5) g 6.246(4) g 6.263(2) 6.255() 6.244(9) g 6.242() g (3) f (9) (6) 6.329(6) (9) 6.329(24) (9) (7) (2) () 6.58(6) (2,3,4) (9) (7) 6.66(4) () 6.66(9) 6.653(25) 6.6() 6.65(7) (3 ) +T (2) (4) 6.77(5) 6.77(2) 6.767(2) 6.766(2) (7) 6.769(2) (3 ) 6.876(2) (9) () (?) (2) - 3 +T 7.27(9) (9) (7) (7) (7) (7) (8) (8) () (6) 7.2(2) 7.69() 7.26(9) (9) (3) (3) (2) (9) (3) (7) (29) () (9) (27) (8) (8) (2) () (8) (9) (5) (4) (4) (6) (6) 7.938(9) (3) - 3 +T (6) (6) Energy from Ref. [6] used for calibration in the present work. a Adopted J π values by Ref. [6]. b Mirror assignment, Fig c Weighted average. d Only data from Refs. [6, 8, 52] are included. e Only present data considered. f Included data in the weighted average are from Refs. [2, 6, 52] g Unresolved doublet. T Theoretical value taken from Ref. [6] Mg data and their astrophysical implications

78 5.4. Region above the proton-emission threshold (5.542 MeV MeV) Mg(p,t) 22Mg Figure 5.6: Doublet near 6.6 MeV observed at the 8 spectrometer angle. The upper panel shows the two-peak fit to the data. The lower panel shows the difference between data and fit (residue spectrum). The determined excitation energies for 22 Mg are listed in the third column of Table 5.2. See further Fig. 4.5 for more details (7) MeV ( ), 6.62(9) MeV (2 + ): This doublet was resolved by Ruiz et al. [9] at 6.587() MeV and 6.6() MeV, respectively. We could resolve these two levels only at an angle of 8 (see Fig. 5.6). At.3 and 7 these lines were affected by the 2 C and 6 O contaminant lines, respectively. By mirror assignments these levels were assumed to correspond to the 6.69 MeV and the 6.89 MeV 2 + levels in 22 Ne, respectively. This is in agreement with Ref. [9] (2) MeV ( +, + ): This level was observed in many previous experiments. Our result is in agreement with all previous results, except that of Ref. [9]. This level is proposed to have spin-parity 3 in Ref. []. In contrast, the mirror state in 22 Ne could be the 6.9 MeV level which has (,) + assignment [6]. In the 2 Na(p,γ) 22 Mg reaction-rate calculations we will assume the natural + spin-parity value. 7.27(9) MeV (3 + )T, 7.45(7) MeV (4 + ), 7.6(7) MeV (3 ), 7.79(8) MeV ( ): These four levels are for the first time observed in our high-resolution (p,t) experiment. We have been able to resolve these levels at spectrometer angles of 8 and 7 (Fig. 5.7). At the spectrometer angle of 8 we resolved all four levels, for 7 only three. These four levels correspond probably to the closely-spaced levels above MeV in 22 Ne (see Fig.

79 Mg data and their astrophysical implications 5.5). We exclude the mirror level at MeV with a probable 5 + spin-parity, since this state is unlikely to be excited in our reaction due to high spin and unnatural parity () MeV + : The + level at MeV is the highest lying level in 22 Mg for which adopted values of spins and parities of excited states are known from other experiments; see Ref. []. Therefore, it is the highest energy where we can verify our spin-parity assignments with 22 Ne mirror states. In this region above the proton-emission threshold and below the alpha-emission threshold we observed additional levels at 7.338(3) MeV and 7.926(5) MeV. Their possible mirror spin-parity assignment can be seen on the right panel in Fig. 5.5.

80 5.4. Region above the proton-emission threshold (5.542 MeV MeV) 73 (a) 7.27(7) 7.45(6) 7.6(6) 7.79(6) 24 Mg(p,t) 22Mg Θ=8 o (b) 24 Mg(p,t) 22 Mg 7.4(3) 7.67() 7.92() Θ=7 o Figure 5.7: Levels observed in the region MeV. Upper panel: spectrum taken at a spectrometer angle of 8 with a 4-level fit and the residue spectrum below it. Bottom panel: spectrum taken at a spectrometer angle of 7 with a 3-level fit and the residue spectrum below it.