Studying the Fusion Evaporation Reaction (α,n) with 54 Fe, 56 Fe, 57 Fe, and 58 Fe. A dissertation presented to. the faculty of

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1 Studying the Fusion Evaporation Reaction (α,n) with 54 Fe, 56 Fe, 57 Fe, and 58 Fe A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Nadyah L. Alanazi May Nadyah L. Alanazi. All Rights Reserved.

2 2 This dissertation titled Studying the Fusion Evaporation Reaction (α,n) with 54 Fe, 56 Fe, 57 Fe, and 58 Fe by NADYAH L. ALANAZI has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by Alexander Voinov Research Assistant Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences

3 3 ABSTRACT ALANAZI, NADYAH L., Ph.D., May 2018, Physics and Astronomy Studying the Fusion Evaporation Reaction (α,n) with 54 Fe, 56 Fe, 57 Fe, and 58 Fe (132 pp.) Director of Dissertation: Alexander Voinov Accurate calculations of nuclear reaction cross sections are in high demand in different areas including basic nuclear physics, astrophysics, and reactor technology. Fusion evaporation reactions are the main type of reactions occurring at low energies of interacting nuclei. The Hauser-Feshbach (HF) theory, based on compound reaction mechanism, is the main tool used for cross section calculations of this type of reactions. It is generally understood that the main uncertainty of calculations is due to uncertainties in input nuclear structure parameters used as inputs in HF calculations. The most uncertain one is the nuclear level density and, more specifically, the spin cutoff parameter determining the spin distribution of the nuclear level density. In this work, the measurements of neutron spectra and neutron angular distributions from 54,56,57,58 Fe(α,n) 57,59,60,61 Ni reactions were performed to study the spin cutoff parameter for 57,59,60,61 Ni isotopes. Experiments were conducted on the tandem accelerator of the Edwards Laboratory using 13.5 MeV alpha beam and the swinger time of flight facility to detect neutrons. The experimental anisotropy of the symmetric neutron angular distributions from compound reactions was used to test different model predictions of the spin cut-off parameter. It was found that the spin cutoff parameter tends to be lower at low excitation energies compared to most popular level density models using, so called, the rigid body estimate.

4 4 DEDICATION To my parents, my husband, and my kids

5 5 ACKNOWLEDGMENTS I would like to sincerely express my gratitude and appreciation to my advisor, Alexander Voinov, for his effective support, personal assistance, and guidance throughout a long way until my dissertation is complete. I would like to thank Prof. Steven Grimes for his valuable knowledge and fruitful discussions. I am also grateful to my dissertation committee members: Dr. Carl Brune, Dr. Alexander Neiman, and Dr. Douglas Green for their efforts of reading the dissertation and giving comments and suggestions to help me to improve my project. My other thanks go to the technical assistance offered to me by Devon Jacobs, Don Carter, and Dr. Tom Massey throughout conducting my experiments at the Edwards Accelerator Laboratory. Particularly, I want to extend my gratitude to Andrea Richard, Cody Parker, Shamim Akhtar, Sushil Dhakal, and Rekam Giri for their assistance in operating the accelerator. I wish to appreciate the financial aid received from King Saud University (KSU). My special appreciation goes to my friends and classmates for their helping and encouragement. Finally, and most importantly, I want to appreciate my supportive husband (Abdulrazzaq Alanazi), my daughters (Aryam, Alaa, Afrah), and my son (Yousef) who provided unending support and inspiration, thank you for your love, patience, and trust that they gave me along these years.

6 6 TABLE OF CONTENTS Page Abstract... 3 Dedication... 4 Acknowledgments... 5 List of tables... 9 List of figures Introduction Objective of the study Thesis outline Theoretical background Nuclear reaction mechanisms Hauser-Feshbach theory Applications of Hauser-Feshbach calculations Nuclear level density Level density models Fermi gas model Constant temperature model Gilbert and Cameron model Microscopical models Fermi gas model parameters Pairing energy parameter Level density parameter a Spin cut-off parameter σ Systematics of nuclear level density parameters Rohr systematics Al-Quraishi systematics T. von Egidy and D. Bucurescu systematics (2005) T. von Egidy and D. Bucurescu systematics (2009) Arthur systematics Optical model potentials Optical model systematics Angular distributions in compound reactions... 37

7 HF2002 code EMPIRE code Experiment details Facility overview Edwards accelerator laboratory Swinger facility Experiment overview Time-of-flight tunnel NE213 neutron detector Stilbene detector Electronics setup and data acquisition system (DAQ) Time of flight method Targets Neutron-gamma identification Time and energy calibration Efficiency calibration Experimental differential cross section Data analysis and results Experimental results Neutron energy spectra Angular distributions Theoretical calculations Energy spectra of outgoing neutrons Angular distributions Comparison between experimental results and theoretical calculations Fe(α,n) reaction Fe(α,n) reaction Fe(α,n) reaction Fe(α,n) reaction Results and discussion Analysis based on χ " Analysis based on anisotropy of angular distributions Summary and conclusions... 98

8 8 5.1 outlook References Appendix A: Experimental differential cross sections Appendix B: Optical model parameters

9 9 LIST OF TABLES Table Page 3.1 Details about the targets Q-values for neutron channel Excitation energies Umax up to which the level scheme is considered to be complete Parameters for the Fermi gas (FGM) level density model from Von Egidy systematics (2005) [44] and Von Egidy systematics (2009) [48] used in our calculations Gilbert-Cameron level density parameters from Arthur systematics [50] used in our calculations Parameters extracted from discrete levels (spin cut-off parameter σdis at the average energy E$) [5] The χ 2 values from neutron angular distributions of 54 Fe(α,n) reaction. En is the energy interval for emitted neutrons, and Ex is the corresponding excitation energy interval for the residual 57 Ni nucleus. First two intervals were excluded from analysis because of strongly asymmetric angular distributions R values from angular distributions of 54 Fe(α,n) for different energy intervals. En is the energy interval for emitted neutrons, and Ex is the corresponding excitation energy interval for the residual 57 Ni nucleus The χ 2 values for 56 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual nucleus 59 Ni R values for angular distributions from 56 Fe(α,n) reaction for different energy intervals. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual nucleus 59 Ni The χ 2 values of the 57 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 60 Ni nucleus R values for angular distributions from 57 Fe(α,n) for different energy intervals. En

10 10 is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 60 Ni nucleus The χ 2 values of 58 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 61 Ni nucleus R values for angular distributions from 58 Fe(α,n) reaction for different energy intervals. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 61 Ni nucleus P-values for 54 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 57 Ni nucleus P-values for 56 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 59 Ni nucleus P-values for 57 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 60 Ni nucleus P-values for 58 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 61 Ni nucleus. First two intervals were excluded because of experimental angular distributions are asymmetric Ranys for 54 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy Ranys for 56 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy Ranys for 57 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy Ranys for 58 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy.. 97

11 11 A.1 Numerical values for the experimental cross sections for outgoing neutrons from the 54 Fe(α,n) reaction at Eα = 13.5 MeV A.2 Numerical values of the experimental cross sections for outgoing neutrons from the 56 Fe(α,n) reaction at Eα = 13.5 MeV A.3 Numerical values of the experimental cross sections for outgoing neutrons from the 57 Fe(α,n) reaction at Eα = 13.5 MeV A.4 Numerical values of the experimental cross sections for outgoing neutrons from the 58 Fe(α,n) reaction at Eα = 13.5 MeV B.1 Optical model parameters for α-particles on target nuclei 54 Fe, 56 Fe, 57 Fe, and 58 Fe at Eα = 13.5 MeV. The potential depths are given in MeV and the radial parameters in fm [52] B.2 Optical model parameters for neutrons in exit channels n+ 57 Ni, n+ 59 Ni, n+ 60 Ni, and n+ 61 Ni. The potential depths are given in MeV and the radial parameters in fm [53]

12 12 LIST OF FIGURES Figure Page 2.1 Formation and decay of compound nucleus C*, where Ec is the excitation energy of the compound nucleus. The figure is taken from Ref. [12] The energy spectrum of a nuclear reaction A(a,b)B. The figure is taken from Ref. [14] Schematic diagram of the tandem accelerator of Edwards Accelerator Laboratory [60] The swinger facility at the Edward Accelerator Laboratory Schematic layout of the swinger, time of flight tunnel, and the NE213 detector The NE213 neutron detector inside the tunnel The stilbene detector is attached to the Swinger at 45 o with respect to the incident beam direction The electronics diagram for the experiment. Diagram courtesy of T. N. Massey Targets holder inside the scattering chamber Counts as a function of the PSD signal for the reaction of 56 Fe(α,n) measured at the angle of 90 o. Units on x-axis is a channel number TOF spectra from the 27 Al(d,n) 28 Si reaction measured at 120 o Time calibration spectrum from HP 5359A delay generator Time of flight calibrated spectra for the 27 Al(d,n) 28 Si reaction measured at 120 o The energy spectrum of emitted neutrons from the reaction 27 Al(d,n) measured at 120 degrees Neutron spectra measured at different angles in Lab frame for 54 Fe(α,n), 56 Fe(α,n) (Top) and for 57 Fe(α,n), 58 Fe(α,n) (bottom) reactions The standard spectrum for the reaction 27 Al(d,n) at 120 deg and Ed=7.44 MeV [62] Efficiency for NE213 neutron detector.. 55

13 Experimental cross sections of neutrons in CM frame measured at angles 135 (left) and 146 (right) with 13.5 MeV alpha beam from 56 Fe(α,n) reaction Experimental angular distributions of emitted neutrons from 54 Fe(α,n) 57 Ni, 56 Fe(α,n) 59 Ni (Top), and 57 Fe(α,n) 60 Ni, 58 Fe(α,n) 61 Ni (bottom) reactions with 13.5 MeV alpha beam. The cross sections have been multiplied by 4π Spin cut-off parameter as function of excitation energy of 57 Ni, 59 Ni (top), and of 60 Ni, 61 Ni (bottom) nuclei. The black point is the experimental spin cut-off derived from discrete levels Level densities of 57,59 Ni (top) and 60,61 Ni (bottom). Solid lines are calculations with FGM (Egidy2005 sys & Egidy2009 sys) and GCM (Arthur sys). Histogram is the density of discrete levels Angular distributions based on Egidy 2005 and Egidy 2009 systematics from 56 Fe(α,n) reaction in two different excitation energy intervals. The cross sections have been multiplied by 4π to convert the units from (mb/sr) to (mb) Energy spectrum from 54 Fe(α,n) reaction. The points are the experimental data and the solid lines indicate the HF calculations with Egidy 2005 and Egidy 2009 models and Empire calculations with Arthur model Neutron angular distributions from 54 Fe(α,n) reaction for different energy intervals. The points are the experimental data and the solid lines are calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π Neutron energy spectrum from 56 Fe(α,n) reaction. The points are experimental data and the solid lines indicate the HF theoretical calculations using models Neutron angular distributions from 56 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π Neutron energy spectrum from 57 Fe(α,n) reaction. The points are the experimental

14 14 data and the solid lines indicate theoretical calculations using different level density models Neutron angular distributions from the 57 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are HF calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections were multiplied by 4π Neutron energy spectrum from 58 Fe(α,n) reaction. The points are the experimental data and the solid lines indicate the HF calculations using FGM and GCM models with different systematics Neutron angular distributions from 58 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are HF calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π Probability versus chi-square for different values of degrees of freedom. The figure is taken from Ref. [66] Anisotropy ratio Ranys as a function of excitation energy from the reaction 54 Fe(α,n) Anisotropy ratio Ranys as a function of excitation energy from the reaction 56 Fe(α,n) Anisotropy ratio Ranys as a function of excitation energy from the reaction 57 Fe(α,n) Anisotropy ratio Ranys as a function of excitation energy from the reaction 58 Fe(α,n) A.1 Neutron differential cross sections of 54 Fe(α,n) reaction for Eα = 13.5 MeV. 105 A.2 Neutron differential cross sections of 56 Fe(α,n) reaction for Eα = 13.5 MeV. 110 A.3 Neutron differential cross sections of 57 Fe(α,n) reaction for Eα = 13.5 MeV. 115 A.4 Neutron differential cross sections of 58 Fe(α,n) reaction for Eα = 13.5

15 MeV

16 16 1 INTRODUCTION Reaction cross sections are needed in many nuclear applications, such as astrophysics (to determine reaction rates which are used to understand the nucleosynthesis and the energy generation in stars), reactor physics, and medical research. Because of the importance of cross sections, theoretical calculations for different reactions are required when experimental data are not available. Because existing theoretical models are mostly phenomenological, i.e. they require input nuclear structure parameters obtained from experiments, the experimental study of these parameters is necessary to reliably predict reaction cross sections. In 1936, Bohr [1] introduced the compound nuclear model. At relatively low incident energies (< ~5 MeV/A), the collision between a projectile and a target nucleus results in the formation of a relatively long-lived compound nucleus, which decays by ejection of a particle, or by emission of gamma rays. The memory about the formation mode is lost except for the conserved quantities such as energy, angular momentum, and parity, so the angular distribution of outgoing particles becomes symmetric about 90 degrees. If a large number of final levels is involved, the statistical approach is suitable to describe compound nuclear reactions. In this case, the statistical Hauser-Feshbach (HF) theory [2] is used for cross section calculations. Theoretical calculations based on the HF model are widely used in astrophysics, basic nuclear physics, and nuclear data evaluations. Important nuclear structure parameters used as inputs for the HF model are transmission coefficients of incoming and outgoing particles as well as level densities of residual nuclei. The transmission coefficients are determined from optical model potentials based on experimental data of elastic scattering and total cross sections [3]. They are obtained with uncertainties usually not exceeding 15-20%. In contrast, the level density is more uncertain above the region of discrete low-lying levels because it is not possible to count them at higher energies due to high density of levels. At this point the level density theoretical models are used. Parameters of level density models currently used in modern reaction codes are obtained mainly from a fit of model analytical formulas to the data on neutron resonance spacing, which are available for many nuclei at

17 17 the neutron separation energy [4], and the density of low lying discrete levels [5]. Analytical functions are normally based on the Fermi-gas model [6] and/or the Gilbert- Cameron model [7] with adjustable parameters (level density parameter a, spin cut-off parameter σ, and the pairing factor) to fit experimental data. However, such a method contains uncertainties associated with unknown spin and parity distributions which are generally not available experimentally. Hence, more experimental data and theoretical developments are needed in the energy region above discrete levels. The spin dependence of the nuclear level density is very important, and it is currently considered to be most uncertain quantity of level density models which affects cross section calculations. It is determined by the spin cut-off parameter, σ, which characterizes the level density spin distribution in excited nuclei. It has been shown in Ref. [8] that the spin cut-off factor can be obtained experimentally from the angular distribution of (α, n) reactions. The angular anisotropy of evaporated particles from compound reactions is due to alignment of orbital momentum of compound nuclei perpendicular to the beam direction due to angular momentum conservation. In this case the magnitude of the angular anisotropy is determined by angular momentum of outgoing particles being completely isotropic if the angular momentum is zero. The anisotropy depends on the energy and mass of the incoming and outgoing particles, and on the spin cut-off parameter σ of residual nuclei, so any changes in σ can affect the cross sections and angular distributions. Therefore, the anisotropy measurements in compound reactions are important to study the spin distribution of levels in residual nuclei [9]. In this work, the spin cut-off parameter was studied experimentally for several residual nuclei 57 Ni, 59 Ni, 60 Ni, and 61 Ni from the compound nuclear reactions α+ 54 Fe, α+ 56 Fe, α+ 57 Fe, and α+ 58 Fe by measuring the angular and energy distributions of outgoing neutrons. The neutron spectra are most suitable for studies of the spin cut-off and the level density for two reasons: neutron transmission coefficients are well known, and the neutron channel is a preferred decay channel of compound nuclei. 1.1 Objective of the study The aim of this work is to study the spin cut-off parameter and level densities for 57,59,60,61 Ni isotopes from neutron evaporation spectra of (α, n) reaction with 13.5 MeV

18 18 alpha beam induced on 54,56,57,58 Fe. The alpha beam is produced by the tandem accelerator of the Edwards Laboratory. It is important to study the dependence of the level density on the neutron number of a specific element; in our case it is Ni. The study will be based on measuring the magnitude of the anisotropy of a symmetric angular distribution of neutrons with the swinger neutron time of flight facility of the Edwards Laboratory. According to Ref. [9], the magnitude of the anisotropy is affected by the spin cut-off parameter of final nuclei (Ni isotopes, in our case). The spin cut-off parameter determines the level density spin dependence according to Eqn. (2.5). Three level density models, which describe different analytical functions to calculate the spin cut-off parameter, will be tested. 1.2 Thesis outline The dissertation is divided into five chapters. In Chapter 2, the theoretical background is intended to provide fundamental concepts in this study. First, the nuclear reaction mechanisms are described. Then, the Hauser-Feshbach model and its applications are presented. The nuclear level density is introduced which is followed by a review of different level density models that are utilized in data analysis. Chapter 3 explains the experimental setup for this study. In Chapter 4, data analysis and results are presented and discussed. Lastly, the summary and conclusions are described in Chapter 5.

19 19 2 THEORETICAL BACKGROUND 2.1 Nuclear reaction mechanisms When a projectile interacts with a target nucleus, two things may happen. First, the projectile may be deflected by the target nucleus potential and change its direction, but it does not lose its energy. This reaction is called elastic scattering. Second, the projectile may interact with the target nucleus and cause a nuclear reaction which can occur mainly via three different reaction mechanisms based on the time of interaction between the projectile and the target; direct, compound, and pre-equilibrium [10]. The direct reaction takes place when a projectile interacts with an individual nucleon in a target nucleus within the time interval it takes for a projectile to pass through a target nucleus. The typical reaction time, which is approximately equal to seconds, can be expressed as 2R/v, where R is the radius of the nucleus and v is the projectile velocity [11]. Since the direct reaction is mostly a single step interaction when the energy of a projectile is not distributed among nucleons in a nucleus, the ejectile remembers the initial direction of the projectile resulting in an angular distribution of ejectiles to be strongly forward peaked [10]. The compound nuclear reactions are important source of information about nuclear structure, and they are represented as, a + A C * b + B, where a, A, C, b, and B are the projectile, the target nucleus, the compound nucleus, the ejectile, and the residual nucleus, respectively. The projectile a is absorbed by the target nucleus A forming the intermediate compound nucleus C *. The energy of the projectile is shared evenly through interaction with constituent nucleons; so all memory about formation mode is lost. The compound nucleus decays then by particle or gamma emission. Thus, the processes of formation and decay are independent of each other [12]. The relatively long reaction time (~10-15 s), which is several orders of magnitude longer than the direct reaction makes the processes extremely slow. Compound reactions occur at relatively low incident energies (< ~5 MeV/A). The particle energy spectra consist of resolved peaks corresponding to particular discrete final states and unresolved distribution corresponding to continuum unresolved states as shown in Fig The pre-equilibrium reaction is a reaction type that takes place before the

20 20 compound nucleus is reached its equilibrium and after the first stage of reaction. The projectile gradually creates more complex states of compound nucleus and also gradually loses its memory. In general, the angular distribution of the emitted particles is forward peaked, but according to the quantum mechanical description of pre-equilibrium reactions, the angular distribution consists of two components; the smooth forwardpeaked component from multi-step direct process (MSD), and the symmetric component from multi-step compound process (MSC) [13]. It can be seen from Fig. 2.2 that the MSD reaction represents the largest part in the pre-equilibrium cross section because the MSC reaction is restricted to energies of MeV [14]. Figure 2.1: Formation and decay of compound nucleus C *, where Ec is the excitation energy of the compound nucleus. The figure is taken from Ref. [12].

21 21 Figure 2.2: The energy spectrum of a nuclear reaction A(a,b)B. The figure is taken from Ref. [14]. 2.2 Hauser-Feshbach theory The compound reactions are described in the framework of the Hauser-Feshbach (HF) theory of nuclear reactions [2]. HF theory is development of the statistical Weisskopf-Ewing theory [15] which was originally proposed to describe particles evaporation spectra from compound nuclear reactions and it does not consider the angular momentum conservation. In contrast, the angular momentum and parity are conserved for the compound reaction in the Hauser-Feshbach model. In this theory, the cross section for the reaction A(a,b)B is determined by transmission coefficients of incoming and outgoing particles and level density of residual nucleus. In this process, the compound nucleus is formed with excitation energy U, angular momentum J, and parity π, which are conserved and determined as U = ε 2 + B 2 = ε 5 + E 5 + B 5, J = i 2 + I 2 + l 2 = i 5 + I 5 + l 5, and π = π 2. Π 2. ( = π 5. Π 5. ( 1)? A where ε 2, ε 5, and B 2, B 5 are the center of mass energies and binding energies of incoming and outgoing particles, respectively; i 2, π 2, I 2, Π 2, i 5, π 5 and I 5, Π 5 are angular momenta and spins for projectile, target, ejectile, and residual nucleus, respectively; l is the orbital angular

22 momentum, and E 5 is the energy of residual nucleus. The compound reaction cross section is given by the following expression σ 2,5 (ε 2, I 2, Π 2 ; E 5, I 5, π 5 ) = σ 2 (ε 2, I 2, Π 2, U, J, π) E A(F,G,H,I A,J A,H A ) G,H. E(F,G,H) 22 (2.1) Here Γ 5 (U, J, π, E 5, I 5, π 5 ) is the decay width for compound nucleus into the state (E 5, I 5, π 5 ) of the residual nucleus by emission b particle, and the cross section for formation of compound nucleus σ 2 (ε 2, I 2, π 2, U, J, π) can be expressed in terms of the transmission coefficients for incoming particles T 2 as ("GOP) σ 2 (ε 2, I 2, Π 2, U, J, π) = H M OP)? 2 (ε where ka is the wavelength number in the incident particle, ("GOP) OP) (2.2) is the statistical factor, and the function is introduced to conserve the parity, it is equal to 1 if the parity is conserved and 0 otherwise. The total decay width Γ is the sum of all partial decay widths (Γ = Γa + Γb + Γc +...) of all outgoing channels. The decay widths can be calculated in terms of the transmission coefficients as P J A OQ A J A TQ A, Γ 5 (U, J, π, E 5, I 5, Π 5 ) = T? A " H U(F,G,H)? 5 (U E 5 B 5 ) f? A A(π) where ρ(u, J, π) is the level density of compound nucleus. (2.3) In general, each nucleus has discrete states at low excitation energies with quantum numbers (E M, I M, Π M ) and continuum levels described by ρ(e 2, I 2, Π 2 ) at higher excitation energies, so in calculations, the total decay width is given by Γ(U, J, π) = 2 Γ 2`(U, J, π, E 2, I 2, Π 2 ). ρ(u 2, I 2, π 2 ) de I [ I + [ M^_ Γ 2`(U, J, π, E M, I M, Π M ), (2.4) where E` is the lower excitation energy of continuum region. All possible emission particles from compound nucleus are included in the summation over a`[16].

23 Applications of Hauser-Feshbach calculations Hauser-Feshbach model that requires knowledge of the nuclear level density plays an important role in various applications such as calculation of reaction rates in astrophysics and cross section evaluations for nuclear technology, specifically, for nuclear data libraries used for designing advanced nuclear reactors. In nuclear astrophysics, calculations of reaction rates for nuclei off of the stability line, especially for neutron-rich nuclei involved in the r-process nucleosynthesis are needed. Explosive nuclear burning in supernovae produces very large number of stable and unstable nuclei, which are not fully investigated experimentally. Thus, it is necessary to estimate reaction cross sections and thermonuclear rates with theoretical models [17]. Many astrophysical calculations for nuclear reaction rates such as the s- or r-process neutron capture are performed based on the Hauser Feshbach statistical model that uses the nuclear level density as an input [2]. In reactor physics, it is important to know precisely the nuclear cross sections with neutrons over large range of energies for large number of nuclei, so that the core reaction rates can be computed accurately. Cross section evaluations as well as their uncertainties are compiled in data libraries such as ENDF/B-VII.1 (the last version) [18]. Evaluation procedures and uncertainty estimations (covariances) are based on available experimental data and model calculations using nuclear reaction codes such as Empire [19]. However, because the large source of uncertainties in theoretical cross section is due to uncertainties in level densities, experimental study of the level density is very important for development of nuclear data libraries used in reactor physics simulations. 2.4 Nuclear level density Nuclear level density is defined as the number of levels per unit of energy at a certain excitation energy. The nuclear levels can be roughly divided into two energy regions based on the approach which are used to determine the level density; low energy and high energy excitation regions. The levels in low energy region are well separated, so the level density can be calculated by level counting. The spacing between levels gradually decreases with increasing the excitation energy. At higher excitation energy region the level density is defined by the level density function ρ(e, J, π), determined

24 indirectly from experiments or from theoretical models [20]. Phenomenological models are used to predict the nuclear density over a certain excitation energy when the experimental information is scarce. The level density function ρ(e, J, π) of specified energy E, spin J and parity π has the following general form ρ(e, J, π) = P(E, π) ρ(e) f(j, E), (2.5) where P(E,π) is the parity distribution, and it is defined as a the ratio of the number of the positive parity states (n+) and number of the negative parity states (n-) at a certain excitation energy E: P(E, π) = 24 c d (I) c d (I) O c e (I). (2.6) In most models, the dependence of the level density on parity π is assumed to be P(E,π)=½ and independent of excitation energy. However, theoretical studies [21, 22], and experimental measurements [23] have shown that levels with one parity type dominates at low excitation energies, while at higher excitation energies, close to the particle separation energy, the positive and negative parity states have equal densities, so the levels approach to an equal parity distribution. The spin distribution function, f(j, E), is generally given by the formula [24] f(j, E) "GOP "g N (I) (GOP/")N exp [ ], (2.7) "g N (I) where σ(e) is the spin cut-off parameter, and it is energy dependent. More discussion about the spin cut-off factor is in section The total level density ρ(e) uses several functional forms for its calculations. The three frequently used semi-empirical analytical formulas are based on the Fermi gas model [6], the constant temperature model [25], and the Gilbert-Cameron model [7] which are discussed in the following section. 2.5 Level density models Fermi gas model Bethe described the nuclear level density based on the Fermi gas model in 1936 [6]. In the Fermi gas model, it is assumed that the nucleons in a nucleus are noninteracting fermions, and they can only occupy single particle states with definite

25 excitation energies, spin projections and parities due to the Pauli exclusion principle. In addition, it is assumed that the single particle states are equally spaced, and the shell effects are not included in this model. Although this model contains a limited physical information, it had been widely using in data analysis. In contrast, many enhanced and modern methods such as Ignatyuk model [26] were introduced to calculate the level density more accurately by considering such nuclear properties as shell effects, pairing effects, and collective excitations. In the Bethe Fermi-gas model, the level density as a function of excitation energy E is given by the formula ρ(e) = 25 nop [" 2I], (2.8) P" " g 2 r/s It/s where σ is the spin cut-off parameter, and a is the level density parameter. The shifted form of the Bethe Fermi gas formula [6], to take into account the even-odd effect in nuclei, introduces the empirical quantity Δ, which is defined as the energy shift. It accounts also for the fact that the nucleon pair in the ground state must be broken with extra amount of energy before each nucleon can be excited individually. Hence, the effective excitation energy U is given by U = E Δ, (2.9) The expression for the total back-shifted Fermi gas model level density with adjustable parameters a, σ and Δ is given by ρ(e) = nop ["w2(itx)]. (2.10) P" "g2 r/s (ITx) t/s In addition, the effective excitation energy can be expressed in terms of the nuclear temperature T as U = at ", so the level density is given by ρ(t) = Constant temperature model nop ("2y). (2.11) P" "g2 z/n yt/n The cumulative number of levels in the energy range up to E is given by the following exponential equation N(E) = exp [(E E _ )/T], (2.12)

26 where T is the nuclear temperature. N(E) is related to the nuclear level density ρ(e) by the formula 26 ρ(e) = }(I) = P exp I y [(ITI ~) ]. (2.13) y In the constant temperature model [7] the nuclear temperature does not depend on the excitation energy, and it is assumed to be constant over entire energy range. The two parameters E _ and T can be determined from fitting the experimental data [25]. The constant temperature model describes the level densities in low excitation energy range typically up to the neutron separation energy Bn [7]. In contrast, the nuclear temperature in Fermi gas model, Eqn. (2.10), varies with the excitation energy as T α E 1/2, and it is given by T = w(e Δ)/a (also see Eq. 2.11) Gilbert and Cameron model In the Gilbert-Cameron model [7], both the constant temperature model (CTM) and the Fermi gas model (FGM) are used to describe the level density in a wide excitation energy region. Gilbert and Cameron found that at the low excitation energy, the nuclear level density is described by the constant temperature model, while at high excitation energy region, the Fermi gas model is better to represent the nuclear level densities. These two models are smoothly connected at the certain matching energy Em. The matching energy Em can be determined from the continuity of the level density function and its first derivative at Em, which leads to the following two conditions and ρ (E ) = P y exp [(E E _)/T], (2.14)?cU ƒ I I^I = P y. (2.15) Hence, the Gilbert-Cameron model requires five parameters to calculate the level density which are the level density parameter a, the pairing energy Δ, the nuclear temperature T, the energy shift Eo, and the matching energy Em Microscopical models Microscopical models is an alternative approach to calculate the nuclear level density in energy and mass regions where experimental data are not available. It attempts

27 27 to avoid some approximations pertaining to phenomenological models such as simplified approach to calculation of pairing, shell and collective effects. Microscopical models use single particle level scheme calculated from realistic effective single particle potentials. They also take into account pairing effects. Several microscopical approaches have been developed in the past decades [27-32] including combinatorial [33], spectral distribution [34], Monte-Carlo [35] and quantum Monte-Carlo [36, 37] models). Although microscopical approaches to calculating the level density are intended to be free from empirical parameters, such inputs as a single particle potential, pairing forces, residual nuclear interactions etc. are still uncertain. Therefore, an accuracy of microscopic level density calculations is still subject for further investigations. The microscopic combinatorial approach [33] is based on the Hartree-Fock-BCS (HF-BCS) predictions of ground state structure properties. This approach has been proven to compete with phenomenological models in reproduction of experimental data (mostly neutron resonance spacings), so it can be used for practical applications. Furthermore, this model provides the energy, spin, and parity dependence of the nuclear level density. It also provides level density prediction at low energies, where the statistical approach cannot be applied. In combinatorial method, the single particle state scheme obtained from the axially symmetric Hartree-Fock-Bogoliubov (HFB) model to build up incoherent p-h state densities as a function of the excitation energy, the spin projection, and the parity. Then, the rotational bands are built up consistently to include the collective rotational effects. As a result, the nuclear level densities were obtained to reproduce very well the experimental data, which include both the cumulative number of low-energy levels and the s- and p-wave resonance spacings at the neutron binding energy with degree of accuracy comparable to phenomenological models. In addition, to enhance the reliability of the microscopic predictions of nuclear level densities, the vibrational enhancement factor is incorporated explicitly in the combinatorial approach by introducing the phonon excitations. The results from this model are in good agreement with the experimental data obtained from the particle-γ coincidence analysis of the reactions ( 3 He,αγ) and ( 3 He, 3 He`γ) [33].

28 2.6 Fermi gas model parameters Pairing energy parameter The pairing energy reflects the odd-even effect in nuclei. The nuclei with odd number of protons or neutrons have lower binding energy. The pairing effect can be mimicked as a shifted ground state energy so that the effective excitation energy is introduced as U = E, and the pairing energy is given by the following relations δz + δn for even even δz for even Z = δn for even N 0 for odd odd, 28 (2.16) where δ Ž and δ } is the phenomenological correction for even-odd differences for protons and neutrons, respectively. The pairing shift can be obtained from mass-differences of neighboring nuclei. In level density calculations, the average values of the pairing energy can be represented as a function of the mass number A as follows [38] = n P", (2.17) where n=0, 1, and 2 for odd-odd, odd-a, and even-even nuclei, respectively Level density parameter a The level density parameter a is associated with the density of single-particle states at the Fermi energy ε and reflects the properties of the single-particle potential. The level density of the nucleus with A nucleons at excitation energy E and spin j, is determined by assuming that two groups of nucleons, neutrons and protons, are distributed in two sets of states. The spacing of the nucleon states near the Fermi energy is assumed to be constant, and it is gp -1 for protons and gn -1 for neutrons. The level density parameter a is given by the analytical expression as a = HN g, (2.18) where the single-particle level density g is given by g = (g p (ε ) + g c (ε )), with g p (ε ) and g c (ε ) are the proton and neutron single-particle level densities near the Fermi energy, respectively [39].

29 The level density parameter has shown to have the linear dependence of a upon mass the number A [39] as where α is a constant obtained from fitting experimental data Spin cut-off parameter σ 29 a = αa, (2.19) The spin dependence in Fermi gas model (Eqn. 2.5) is determined by the spin cutoff parameter σ, and it characterizes the width of the spin distribution of nuclear levels. There are two approaches to define σ. The first one is the semi-classical approach which assumes the nucleus is a rigid body sphere with the radius R = 1.25A P/ fm and moment of inertia I = ( " )μar", where μ is the nucleon mass, and A is the mass number. This gives the spin cut-off factor as which results in the following formula σ " = J ħ N š(it ) = " (P." ) N A / š (IT ), 2 ħ N 2 (2.20) σ " = A / š (IT ) 2. (2.21) The second approach is the quantum mechanical one obtained in Ref. [24]. Ericson has proved that the spin cut-off parameter is given by σ " = g m " T, (2.22) where g = H N a is the single particle state density, m" is the average of the square of the single particle spin projections, and T = w(e Δ)/a is the nuclear temperature. According to the Gilbert and Cameron [7], the value of m " = A "/, which leads to the following formula for the spin cutoff parameter σ " = H N a (0.146 A"/ ) w(e Δ)/a = A "/ wa(e ), Ref. [40] used the expression for m " = 0.242A "/, which yields (2.23) σ " = A "/ wa(e ). (2.24)

30 2.7 Systematics of nuclear level density parameters The term "systematics" indicates the way to calculate level density parameters for any nucleus at any excitation energy, spin, and parity. The "systematics" implies establishing of regularity over mass number, excitation energy, etc. Practically, most of systematics are based on neutron resonance spacing experimental data. However, each systematics uses different approaches to determining such parameters as level density parameter, the spin cutoff parameter, pairing energies, etc. The following sections discuss some systematics of nuclear level density parameters in details Rohr systematics Rohr [41] established the trend of the level density parameter a with the mass number A. There is overall linear dependence of a upon A, but there are discontinuities of the level density parameter at A=38, 69, 94, and 170. These breaks prevent a representation of the data by a single line in a whole mass range. Alternatively, the linear dependence is given by the following equations where and 30 a = A + VAR, (2.25) 1.64 for A for 38 < A 69 VAR =, 6.68 for 69 < A for 94 < A 170 a = A for A > 170 (2.26) The parameter systematics was established based on neutron resonance data. The spin cut-off parameter was adopted as σ " = A "/ wa(e ), (2.27) and the pairing energy Δ is defined as the sum of neutron and proton pairing energies, P(N) + P(Z) as shown in Ref. [7].

31 Al-Quraishi systematics The work by Al-Quraishi [42] suggested two forms for the level density parameter a. It considered the isospin effect to describe the level density parameter a for nuclei far from the valley of stability. The analysis was based on levels known from discrete level scheme only. Analysis showed that the level density parameter a depends on Z and N, not simply on A. The first form accounts for the effect of isospin conservation in nuclei and the a parameter is given by the expression a = αa/exp [β(n Z) " ], (2.28) where the parameter values are α = and β = The level density parameter a in this form has a maximum value at a given A, when the number of neutron N equals the number of protons Z, N = Z = A/2. The second form states that the level density parameter a decreases for nuclei away from the stability line. This form is given by a = αa/exp [γ(z Z «) " ], (2.29) where α = , γ = and Z «= A/( A "/ ) which is the Z value for the beta stable isotope with the mass number A, and it was obtained from the semi-empirical mass formula for the nuclei without the pairing term. For the parity ratio, Al-Quraishi [43] proposed the following formula P(E) = P " [1 ± P ], (2.30) POnop [`(IT )] where ± sign depends on whether the positive or negative orbit is filling, and it is + at low E for P(0)~1 and - for P(0)~0. The c value is 3 MeV -1 and the energy shift δ p = a _ + a P /A 2 N where the constants a _, a P, and a " depend whether the nucleus is eveneven, even-odd, odd-even, or odd-odd, and their values are obtained in Ref. [43]. The analysis in Ref. [43] has used two forms of the spin cut-off parameter. The first expression is based on the assumption of a rigid body model of spherical nucleus with radius 1.25A 1/3 fm σ " = A / w(e )/a. (2.31) The second one is derived from the statistical calculation of the spin projection on Z-axis

32 32 values averaged over the single particle states. It is given by σ " = A "/ wa(e ). (2.32) T. von Egidy and D. Bucurescu systematics (2005) Von Egidy and Bucurescu [44] determined the empirical level density parameters for the back shifted Fermi gas model (BSFG) and the constant temperature model (CTM) for 310 nuclei by fitting the complete levels at low excitation energies and neutron resonances at the neutron binding energies. For the BSFG model, the proposed formula for the level density parameter a is based on established correlation between a and values of the shell correction S(Z,N), and it is given for all nuclei a/a = p P + p " S (Z, N) + p A, (2.33) where the constants p1, p2, and p3 were found to be 0.127, T, and T, respectively, S (N, Z) = S(N, Z), and the shell correction S(N,Z) is defined as S(N, Z) = M nop M ³, where Mexp is the experimental mass and it is taken from the Audi-Wapstra masses table [45], and MLD is the calculated mass from the macroscopic liquid-drop model [46]. The pairing correction is given with +0.5P for even even = µ 0 for odd A (2.34) 0.5P for odd odd. Here, Pd is the deuteron pairing and it is calculated from the following formula P = P ( 1)ŽOP [S (A + 2, Z + 1) 2S (A, Z) + S (A 2, Z 1)], where Sd is the deuteron separation energy. The formula for the back shifted energy parameter E1 is E P = p»(ž,}) P 0.5P + p»(ž,}) p " 0.5P + p»(ž,}) p + 0.5P + p for even even for odd A for odd odd. (2.35) (2.36)

33 where the fitted parameter values are found to be p P = 0.468, p " = 0.565, p = 0.231, and p = This back shifted parameter E1 has the same physical meaning as in Eqn. (2.10). The derivative of the shell correction with respect to A,»(Ž,}), is calculated to consider some structures in the energy backshifts E1 such as that around mass number 200 due to the shell closure at Z=82. It is calculated as»(ž,}) = [S(Z + 1, N + 1) S(Z 1, N 1)]/4. (2.37) The spin cut-off parameter has a dependence on the level density parameter a and the excitation energy E. It is given by the following expression, which is proposed in Ref. [47] σ " = A / POwPO 2(ITI r). (2.38) " T. von Egidy and D. Bucurescu systematics (2009) In Ref. [48], von Egidy and Bucurescu proposed a new empirical formula for the spin cutoff parameter. The new formula has been exclusively derived from analysis of discrete level scheme in nuclei. No data on neutron resonance spacing were used. They obtained the spin cut-off parameter as a function of mass number A and excitation energy E for 227 nuclei between F and Cf. The resulting formula for the spin cut-off parameter is σ " = 0.391A _. ¼ (E 0.5Pa ) _. P", (2.39) and the deuteron pairing energy Pa is calculated from the formula Pa = P [M(A + 2, Z + 1) 2M(A, Z) + M(A 2, Z 1)], (2.40) " where the mass values M(A, Z) are taken from the table of Ref. [45]. is For the BSFG model, the corresponding formula for the level density parameter a a = ( S )A _.½ ¾, (2.41) where the shell correction is S = S 0.5Pa. The following expression is used for the back shifted energy 33 E P = Pa. (2.42) The basic difference between this approach (von Egidy 2009) and the previous

34 34 approach (von Egidy 2005) is due to the difference in the mass and the energy dependence of the spin cutoff parameter. von Egidy 2009 systematics has an energy dependence of E and mass dependence of about A 0.7 while the von Egidy 2005 systematics has an energy dependence of E 0.5 and A dependence of A 1.2 for spin cutoff parameter Arthur systematics It is followed from neutron resonance data that for nuclei close to the closed shells, the level density parameter a deviates significantly from the expression (2.19). Moreover, nuclear level density shell effects disappear at higher excitation energies. To take both effects into account, Ignatyuk [26] suggested a new expression for a, which has shell and excitation energy dependence, and it is given by the following expression a(u, Z, A) = a (A)[1 + δw(z, N) PTneÁÂ ]. (2.43) Here, a (A) is the asymptotic level density parameter, δw(z, N) is the shell correction which is determined from mass difference between the experimental nuclear mass Mexp and the mass from the liquid drop model calculation MLD, and γ is the shell effects damping parameter. As seen in Eqn. (2.43), the shell effect disappears at high excitation energies (U ) due to damping term, and the level density parameter a(u, Z, A) reaches the asymptotic level density parameter a (A). The damping parameter γ shows how rapidly the damping of the shell effects occurs. The asymptotic level density parameter a (A) and the damping parameter γ are obtained from different systematics such as Ignatyuk [26], Iljinov [49], and Arthur [50] which are given by a (A) = 0.154A T A " and γ = 0.054, (2.44) a (A) = 0.114A T" A "/ and γ = 0.054, (2.45) a (A) = 0.137A T A " and γ = 0.054, (2.46) respectively. These parameter systematics are implemented in Empire reaction code as options [19]. They are used in Gilbert and Cameron model of the nuclear level density. The spin cutoff F

35 35 parameter is determined above the matching energy Em from the expression σ " (E) = A "/ wa(e ), (2.47) and it is given as a linear interpolation between Eqn. (2.47) and the spin cut-off from discrete levels in the energy range below Em. 2.8 Optical model potentials The optical model [5] is widely used in nuclear reaction theory calculations, specifically in obtaining the particle transmission coefficients which are essential ingredients in Hauser-Feshbach statistical theory, as shown in section 2.2. The basic assumption of the optical model is that the interaction of a nucleon with a target nucleus can be replaced by interaction of particle with a potential well. The strength of interaction depends only on the radial distance r between the nucleon and the nucleus. Since the potential is complex, the real part determines the elastic scattering, and the imaginary part is responsible for the absorption of the particle. It is called optical model because the scattering of particles on nuclei is similar to the scattering and absorption of light by a medium. The assumption of the optical model is sufficiently useful only for the energy range from one MeV to several hundred MeV [3]. The optical potential is mostly given by the following expression V(r) = [VV (r)+ VS(r)] + VSO(r) + VCoul(r), (2.48) which consists of three fundamental terms. First term is the sum of the volume VV (r) and the surface VS(r) nuclear potentials with real and imaginary components. The second one is the spin-orbit potential VSO(r) to account for the polarization of the scattered particles. The third term VCoul(r) is the Coulomb potential, and it is added for incident charged particles. The analytical expression for optical potential at a given incident energy is V(r) = V Å f ÆÅ (r) iw Å f ÇÅ (r) + 4ia» W» d dr f Ç»(r) +( ħ É`)" P Ê [V»Ë f Ê Æ»Ë(r) + iw»ë f Ê Ç»Ë(r)]2(SÌ. LÎ) + V Ï (r), (2.49) where VV and WV are the real and imaginary well depths for the volume potential, WS is the imaginary well depths for the surface potential, VSO and WSO are the real and

36 36 imaginary well depths of the spin-orbit potential, mπ is the pion mass, (SÌ. LÎ) is the scalar product of the intrinsic spin SÌ and orbital angular momentum LÎ operators, which is given by for incident nucleons l for j = l + P " 2(SÌ. LÎ) = µ (l + 1) for j = l P, (2.50) " The Coulomb potential V Ï is added for the charged projectiles, and it is given by V Ï (r) = µ ÑŽn N "Ò [ (3 ÊN Ò Ó N) ÑŽn N Ê, r < R`, r R` (2.51) where z and Z are the charge numbers of projectile and target, respectively, and RC is the Coulomb radius of the homogeneous spherical charged nucleus. The geometric factors f Q (r) are usually of Woods-Saxon form, which is given by fi(r) = {1 + exp[(r Ri)/ai]} 1. (2.52) The nuclear radii are defined as Ri = ria 1/3, and ai is the potential diffuseness parameter in fm with i = r, v, s, so, c. Optical model potentials are determined from experimental data on elastic and total reaction cross sections. The library of optical potentials is available for different combination of projectile and nuclei in Ref. [5] Optical model systematics The main principle to obtain the optical model parameters (OMP) for the phenomenological optical potentials is by adjusting these parameters to reproduce the experimental data on elastic scattering and total reaction cross sections over a certain range of incident energies and nuclei. Several systematics have been established for variety of incident particles. For example, OMP of Demetriou et al [51], and Avrigeanu et al [52] are used to determine the global alpha-nucleus optical model potentials. Koning and Delaroche [53], and Soukhovitskii et al [54] parameterizations have obtained for nucleon-nucleus potentials. In our data analysis, OMPs of Avrigeanu et al [52] for α-particles are utilized to

37 calculate the cross sections. Avrigeanu et al obtained these parameters by extending the global optical potentials of Nolte et al [55] to low energies. However, Nolte et al fitted the data of alpha particle scattering with energies more than 80 MeV on six target nuclei 12 C, 40 Ca, 50 Ti, 58 Ni, 90 Zr, and 208 Pb using the Woods-Saxon form factor. These potentials were able to reproduce the experimental data with good accuracy. For protons and neutrons, OMPs of Koning and Delaroche systematics [53] was used in our calculations. These parameters are derived using a smooth function of potential depths with an energy dependence, and constrained geometry parameters for nuclei with mass range and incident energy range 1 kev MeV. These global nucleon potentials were in a good agreement with the experimental scattering data. 2.9 Angular distributions in compound reactions In a compound nuclear process (a + A C * b + B), in order to conserve the angular momentum, the spin of the compound nucleus C * tends to be aligned with the orbital momentum of an incoming particle a. Since it is always perpendicular to the direction of incoming beam, the spin of compound nuclei is aligned perpendicular as well. As a result, the angular distribution of the emitted particles b is anisotropic but symmetric about 90 degrees, and anisotropy is determined by the angular momentum of emitted particle. Because this angular momentum is determined by spin distribution of initial and final levels, the spin cutoff parameter of final levels can be obtained from analysis of the angular distribution of emitted particles [9]. Here we assume that the spin distribution of initial compound levels is obtained accurately from optical model calculations using optical transmission coefficients in the entrance channel. Ericson and Strutinsky [9] have shown that the angular distribution from compound nuclear reaction is approximately given by 37 W(E 5, θ) ~ 1 N? A N "("g N ) N ħ s cos" θ, (2.53) where E 5 is the energy of emitted particles, σ " is the spin cut off factor for the nucleus populated by emitted particle, θ is the CM angle, and l 2 " = μ 2 E 2 and l 5 " = μ 5 E 5 are average of the square of angular momenta of projectile and emitted particle, respectively. μ 2, E 2 and μ 5, E 5 are the masses and energies of the projectile and ejectile, respectively.

38 38 It can be seen from Eqn. (2.53) that the spin cutoff parameter σ " can be determined if the angular distribution is measured experimentally. Average angular momentums are determined from optical model calculations discussed in section 2.8. The larger anisotropy in angular distributions is expected when the angular momenta l 2 and l 5 are larger. Therefore, angular distributions from α -induced reactions are usually studied [8, 56, 57, 58] because alpha particles transfer larger angular momentum compared to nucleon induced reactions. The latter such as (p,p`), (n,n`) and (p,n) have small anisotropy, so they are not useful to determine σ values. It is also found that the anisotropy for α-particle induced reactions decreases as the mass number A increases beyond 100. The formula (2.53) is approximate. The more accurate angular distribution from compound nuclear reactions can be calculated using quantum-mechanical approach which is implemented in the reaction code HF2002 described in the following section HF2002 code HF2002 [59] is a Hauser-Feshbach code written at Ohio University by S. M. Grimes. In this code, the evaporated n, p, d, t, 3 He, α-particles and γ-rays are allowed. It allows a total of ten stages of particle decay and an unlimited number of γ-rays. In addition, the isospin conservation at each reaction stage is included. The Q-values are not read separately because the code has the Wapstra mass table. The continuum level densities with unequal distributed parities are allowed unless they are specified to be equal. The angular distributions are also calculated for the first stage of compound nuclear decay. The input file of the HF2002 code includes information about the compound reaction such as energy of projectile in MeV, mass number and atomic number of first compound nucleus. The entrance channel and the reference frame for transmission coefficients should be specified. Moreover, there are different selections for the input parameters for level density of particular nuclei (a, δ, and σ). The energy value for discrete levels cutoff should be specified, so the continuum level density starts above this energy. The distinctive feature of this code is that the spin cutoff parameter can be input in table form using numerical values as a function of the excitation energy. This allows

39 39 analysis of angular distributions affected by the spin cutoff to be independent from specific functional form of the spin cutoff parameter function σ(e). The code can output calculations for cross sections and angular distributions for different exit channels (neutron, proton, deuteron, triton, helium-3, or alpha) EMPIRE code EMPIRE code [19] is publically available and designed for nuclear reaction calculations. It is also used for cross section evaluations for ENDF nuclear data library. The code accounts for various nuclear reaction mechanisms, which include direct, preequilibrium, and compound reactions. The code works over wide range of incident energies (from ~kev to several hundred MeV) and different projectiles (neutrons, protons, alphas, and photons). There are several options for nuclear level density inputs: Empire-specific level densities, Generalized Superfluid Model (GSM) by Ignatyuk et al [26], the Hartree-Fock-BCS microscopical model (HFB) [33], and Gilbert-Cameron model (GCM) [7]. The library of input parameters includes nuclear masses, optical model parameters, ground state deformations, discrete levels and decay schemes, level densities, fission barriers, moments of inertia and γ-ray strength functions. Both HF2002 and Empire codes were used for analysis of experimental cross sections presented in this work.

40 40 3 EXPERIMENT DETAILS The fusion evaporation reactions (α,n) on 54 Fe, 56 Fe, 57 Fe, and 58 Fe were measured during the experimental campaign conducted at the Edwards Accelerator Laboratory of Ohio University. In this section description of experimental facilities, detection systems and experimental methods will be described. 3.1 Facility overview Edwards accelerator laboratory The 4.5 MV tandem Van de Graaff accelerator in the Edwards accelerator laboratory at Ohio University was used to perform experiments. The lab layout is shown in Fig There are two types of ion sources: the Cs sputter source to produce H, D, and heavy ions, and a Duoplasmatron source to produce 3 He and 4 He. The negative ions that are extracted from the ion source are deflected by the injection magnet and focused by the Einzel lens and XY steerers. The injected negative charged beam is attracted to the positive terminal at high voltage, and accelerated by passing through a carbon foil stripper inside the terminal so that it becomes a positively charged beam. The ion beam is then guided and focused by a series of slits, steerers, and quadrupole magnets. The beam is sent directly to the swinger facility, which is located in the vault area, or to the small or large target rooms by using the analyzing and switching magnets. The 90 analyzing magnet identifies the particles based on their energy, mass, and charge state. The nuclear magnetic resonance (NMR) probe is used to set the magnetic field for beam path control. The switching magnet redirects the beam to five beam lines; three are located in the large target room, and two are in the small target room. The pulsing and bunching system of the accelerator converts the direct current (DC) beam to a pulsed current (AC) beam. Hence, the pulsed beam can be used for neutron measurements with the time-of-flight method [60].

41 41 Figure 3.1: Schematic diagram of the tandem accelerator of Edwards Accelerator Laboratory by Catalin Matei [60] Swinger facility The swinger facility of the Edwards Accelerator Laboratory is used for high precision neutron time-of-flight measurements. It consists of two magnets, which deflect the incoming beam, so the beam comes out from the swinger perpendicular to the original beam path. The swinger can rotate to forward and backward angles between -4 o and 150 o, so the direction of incoming beam and the target rotates with respect to the detector in the tunnel, which allows for a measurement of the angular distribution of the emitted neutrons. The target chamber in this experiment was attached to the end of the swinger as shown in Fig. 3.2.

42 42 Figure 3.2: The swinger facility at the Edward Accelerator Laboratory. 3.2 Experiment overview Time-of-flight tunnel The neutron time-of-flight measurements were carried out using the well-shielded 30 m tunnel, which allows for low background measurements of neutrons. The 1-m-thick concrete walls shield the tunnel from scattered neutrons and γ-rays that are produced in the vault area. The vault area and the tunnel are connected with the collimator, which is a 30 cm diameter cylindrical hole in the wall filled with cylindrical polyethylene sections. Inner diameter of these sections are optimized with experimental geometry to minimize scattered neutrons hitting a neutron detector. After the interaction between beam and a target, the emitted neutrons pass through the collimator and get detected by the NE213 neutron detector which was positioned at the distance of 5.14 m (flight path) from the target. Figure 3.3 shows a schematic layout of the experiment.

43 43 Figure 3.3: Schematic layout of the swinger, time of flight tunnel, and the NE213 detector NE213 neutron detector A NE213 liquid scintillation detector (Fig. 3.4) is an organic scintillator that is used for neutron detection. It has a cm diameter and 5.08 cm thickness, and is coupled to a photomultiplier tube (PMT). The actual total kinetic energy of neutrons cannot be extracted from the pulse amplitude because a neutron does not deposit all its energy in the detector. Therefore, the neutron energy is measured by using the time-offlight (TOF) technique. The output signal of the PMT is a current pulse that is used for getting information about energy deposited in the detector, time it occurred and type of particle detected whether it is a neutron or a gamma ray. Information about the energy deposited by the neutron is determined from the total scintillation light collected by PMT, while the shape of the pulse gives important information about the charged particle that generated it

44 44 (see Sec ). Electrons produce pulses with shorter fall times than the ones produced by protons. Figure 3.4: The NE213 neutron detector inside the tunnel Stilbene detector The stilbene detector (Fig. 3.5) is an organic scintillator detector composed of C14H12 crystals. It is used as a reference for the neutron flux that is produced from beamtarget reactions. This monitor detector is attached to the swinger line in the vault area at fixed angle of 45o with respect to the incident beam. Hence, the data collected by the stilbene detector does not change when the swinger angle changes, so it helps to check if any parameters are changed during the experiment.

45 45 Figure 3.5: The stilbene detector is attached to the Swinger at 45 o with respect to the incident beam direction Electronics setup and data acquisition system (DAQ) The electronics setup for the experiment is shown in Fig The NE213 neutron detector is connected to a photomultiplier tube (PMT) which is set at a high voltage (HV) of V. Photons produced from the scintillating material are guided to the PMT to convert them to an electrical signal which is proportional to the energy deposited in the detector. The signal is split into two components: fast and slow. First, the slow signal from the dynode is linear and it is passed through the Ortec 113 preamplifier and the Ortec 572 amplifier in the tunnel for signal amplification. The output signal is then passed to an analog-to-digital converter (ADC) to produce the energy signal. Second, the fast signal from the anode is split into two signals. One of the signals goes to a Mesytec MPD-4 module, and it is passed through an Ortec 427A delay amplifier and then to an analog-to-digital converter (ADC) for pulse shape discrimination (PSD). The second signal proceeds to a fast amplifier and then to an Ortec 934 discriminator. This output signal is passed to an Ortec 437A time-to-analog converter (TAC) for TOF information.

46 46 Once this process produces the energy signal from the pulse height and real time data, they are collected through the data acquisition software system (DAQ) located in the control room at the Edwards Accelerator Laboratory. These data are acquired in event mode, which means the pulse height and time information are saved for each single event. These events are stored, so they can be replayed for the data analyses later. During each run, event files with assigned number are saved on a computer hard drive for later processing. Figure 3.6: The electronics diagram for the experiment. Diagram courtesy of T. N. Massey Time of flight method The energy of outgoing neutrons is obtained from the time interval needed for a neutron to reach a neutron detector (time of flight). This method uses the detector s fasttiming capability. The time of flight technique requires the beam to be pulsed. In order to

47 produce a pulsed beam, the beam pulsing and bunching method is used with about 3- nanosecond timing resolution. The time difference between start and stop signals is defined as the relative flight time, and is used to extract the energy of neutrons. The beam pick-off (BPO) device produces the start signal, and the neutron detector generates the stop signal when neutrons or γ-rays are registered. The BPO signals undergo a long delay, so they are registered in the DAQ system after the detector signals. Hence, in the time-of-flight spectrum, the channels have a reverse order relative to the absolute time of flight. In order to convert the TOF spectrum to the neutron energy spectrum, the following classical expression of the kinetic energy is used E c = 47 Ø ³ N "y N, (3.1) where L is the flight path, T is the neutron time of flight, and mn is the neutron mass. For γ-rays, we apply the same time-of-flight concept because the speed of light c and the flight path L are constant, so the corresponding flight time is constant too. The position of the γ-peak in time of flight spectrum is utilized for absolute time calibration Targets In this experiment, four foil-type iron targets are used which are 54 Fe, 56 Fe, 57 Fe, and 58 Fe. More details about the targets are in Table 3.1. These targets are fixed on a circular target holder inside a scattering chamber (Fig. 3.7). This chamber is attached at the end of the swinger beam line, and it is under high vacuum. In addition, an Aluminum target is used to measure the standard calibration reaction 27 Al(d,n) for detector efficiency calibration. The contribution from the carbon contamination deposited on iron targets is measured with a carbon target to subtract the 12 C peaks from Fe(α,n) neutron spectra. Finally, the neutron spectra are collected at different angles by using empty target to measure background contributions.

48 48 Table 3.1: Details about the targets. Element Purity (%) Thickness mg/cm 2 54 Fe T 56 Fe T 57 Fe T 58 Fe T Figure 3.7: Targets holder inside the scattering chamber.

49 Neutron-gamma identification Both neutrons and γ-rays are detected in the NE213 liquid scintillator. When the detector absorbs the energy of neutrons, they generate α-particles or recoil protons, while γ-rays generate fast electrons. The electrons are faster and more lightly ionizing than heavy particles, so the light in the scintillator is emitted in approximately two components: fast and slow. The neutron light pulse is longer in time than the light pulse produced by γ-rays. Therefore, the pulse-shape-discrimination (PSD) technique based on analysis of the pulse shape can be used to distinguish between neutrons and gammas in liquid scintillators [60]. The PSD signal is proportional to the fall time of the electrical signal from the neutron detector. The example of the PSD spectrum obtained from the reaction 56 Fe(α,n) is shown in Fig. (3.8). The left peak represents the γ-rays, and the right peak represents the neutrons. Figure 3.8: Counts as a function of the PSD signal for the reaction of 56 Fe(α,n) measured at the angle of 90 o. Units on x-axis is a channel number.

50 Time and energy calibration Both time and energy calibrations of the detector are needed to convert the channel numbers in the TOF spectrum to the corresponding absolute flight time, and then, to the neutron energy. In order to make that calibration, the high yield of neutrons from the 27 Al(d,n) reaction was measured. The time of flight spectra for the emitted neutrons and γ-rays from 27 Al(d,n) 28 Si reaction at 120 o with deuteron energy of 7.44 MeV are plotted in Fig. (3.9). This reaction was measured with the NE213 detector placed at a ~5 m flight path inside the tunnel. It can be seen from Fig. (3.9) that the neutrons (black curve) and γ-rays (red curve) spectra were separated by using PSD technique, but there is a small peak in the neutron spectrum around the channel number of 3500 which is due to the small number of γ-rays misidentified as neutrons. The time calibration is applied by using the following equation TOF = a * channel + b, (3.2) where a is the slope, and b is the intercept. First, the HP 5359A delay generator is used to determine the slope a. The spectrum from the pulse generator (Fig. 3.10) is a series of five peaks that are separated by a specific time interval. In this experiment, the time interval between two neighboring peaks is 50 ns, so the slope a in units of ns/channel is calculated as a=(time interval)/(channel interval). Secondly, the intercept b is determined from the position of the γ-ray peak in Fig. (3.9) which represents the time that requires for γ-ray to reach detector. By knowing the flight path and the speed of light, we can calculate the value of b. Now, by applying the linear equation (3.2), we can calculate the absolute time of flight, and plot the spectra for 27 Al(d,n) reaction, which is shown in Fig. (3.11). Finally, once the absolute TOF spectrum is obtained, the energy of neutrons can be calculated from Equation (3.1) to plot the energy spectrum in Fig. (3.12). Figure 3.13 shows neutron energy spectra in Lab frame at different angles for the reactions 54,56,57,58 Fe(α,n).

51 51 Figure 3.9: TOF spectra from 27 Al(d,n) 28 Si reaction measured at 120 o. Figure 3.10: Time calibration spectrum from HP 5359A delay generator.

52 52 Figure 3.11: Time of flight calibrated spectra for the 27 Al(d,n) 28 Si reaction measured at 120 o. Figure 3.12: The energy spectrum of emitted neutrons from the reaction 27 Al(d,n) measured at 120 degrees.

53 53 Figure 3.13: Neutron spectra measured at different angles in Lab frame for 54 Fe(α,n), 56 Fe(α,n) (Top) and for 57 Fe(α,n), 58 Fe(α,n) (bottom) reactions Efficiency calibration The NE213 detector is not 100% efficient, so the number of neutrons that are registered must be corrected by efficiency of the NE213 detector. The efficiency calibration of the neutron detector was applied by using a standard neutron spectrum measured from the reaction 27 Al(d,n) at 120 deg and Ed=7.44 MeV with a target thick enough to stop the deuteron beam [62]. In Ref [62], the measurements were done using a 235 U foil inside a fission chamber. The uranium isotope fission cross sections are precisely known, so they were used to get the absolute neutron yield which is used as a standard for detector efficiency calibration. Such a standard spectrum is presented in Fig By comparing the experimental spectrum measured with NE213 detector (presented

54 54 in Fig. (3.12)) with the standard spectrum in Fig. (3.14), we can calculate the efficiency of the detector in the energy range En = MeV from the formula Efficiency = Nexp / Nstd, (3.3) where Nexp is the yield for the experimental spectrum per solid angle Ω per integrated charge C, and Nstd is the yield for the standard spectrum per solid angle Ω per integrated charge C. The efficiency of the neutron detector as a function of the neutron energy is shown in Fig. (3.15). The peaks in the efficiency curve are due the interaction between the neutrons and nuclei in air, the aluminum in the scattering chamber, or the carbon in the NE213 detector [63]. Once the detection efficiency is incorporated in the measured cross section, the effect of these interactions is eliminated.

55 55 Figure 3.14: The standard spectrum for the reaction 27 Al(d,n) at 120 deg and Ed=7.44 MeV [62]. Figure 3.15: Efficiency for NE213 neutron detector.

56 Experimental differential cross section In order to compare the experimental spectra with theoretical calculations, the number of particles registered in the detector should be converted to the reaction cross section. The experimental double differential cross section in the lab frame is calculated by N g Û I = 56 } Ü } Ý. Þ. Û. c ß. à. I, (3.4) where NS is the number of scattered particles per energy E and solid angle Ω intervals, Ni is the number of incident particles per energy E and solid angle Ω intervals, and is computed from the integrated beam current (BCI), t is the target thickness, nt is the number of atoms per unit volume of the target, Ω is the solid angle and it is determined from the effective area of the detector and the distance between the detector and the target, and ε is the detector efficiency. In the further analysis, the double differential cross sections ( N g ) are usually multiplied by 4π to be able to compare them with Û I calculations from reaction codes which produce differential cross sections ( g ) in units of mb/mev, i.e. angle integrated cross sections. The possible systematic uncertainties of experimental cross sections from each are roughly the following: about 1% from the BCI counter, 15% from thickness of target, and 5% from detection efficiency, so that the total systematic uncertainty of measured cross sections is approximately 15%. This uncertainty should be applied to the general scaling of differential cross sections, they should not affect uncertainties in shapes of both experimental energy spectra and angular distributions. The experimental differential cross section in the lab frame were transformed to the center-of-mass frame (CM) to compare them with theoretical calculations. More details about experimental neutron cross sections are presented in the next chapter (Sec. 4.1). I

57 4 DATA ANALYSIS AND RESULTS In this chapter, the data analysis of evaporated neutrons and analysis results are presented for the reactions of α + 54,56,57,58 Fe measured with the alpha beam energy of 13.5 MeV. The experiment was carried out at the Edwards Accelerator Laboratory by using the swinger facility. The NE213 neutron detector located at the flight path of 5.14 m inside the well-shielded TOF tunnel was used to detect neutrons. The diameter of the detector is cm, and the thickness is 5.08 cm, so the solid angle subtended by the detector was calculated to be sr. The emitted neutrons were detected at 20 o, 34 o, 45 o, 62 o, 76 o, 90 o, 118 o, 135 o, and 146 o angles for 54,56,58 Fe(α,n) reactions, and at the angles 35 o, 53 o, 72 o, 90 o, 109 o, 128 o, and 146 o for 57 Fe(α,n) reaction. 4.1 Experimental results Neutron energy spectra Neutron differential cross sections 57 N g ( I Ø á) (E c, θ) (also referred to as spectra of evaporated neutrons) were measured from reactions of α+ 54,56,57,58 Fe with the alpha beam energy of 13.5 MeV to study residual nuclei 57,59,60,61 Ni populated by neutrons. Neutron spectra are most suitable for level density studies because neutron transmission coefficients are better known than proton and α-transmission coefficients. In addition, neutrons, in most cases, is a preferred decay channel for compound nuclei because there is no Coulomb barrier. Neutrons can be emitted with low average energies which correspond to population of higher excitation energies of residual nuclei. Also, very small pre-equilibrium fraction is expected to contribute to differential cross sections [56]. TOF spectra were measured and converted to neutron energy spectra. γ-ray events were removed by using the pulse shape discrimination (PSD) technique. Background neutron spectra, which were due to interaction of beam with the beam stop in the chamber and beam collimators were measured separately with a blank target and subtracted. Measured neutron energy spectra were transformed from the lab frame to the center-of-mass (CM) frame to compare with theoretical calculations, since calculations are generally given in the CM frame. Equation (3.4) was applied to calculate the differential cross sections in lab frame of emitted neutrons. The transformation from lab

58 58 frame to CM frame for energy, angel, and differential cross section of outgoing neutrons ã Ø from (α,n) reaction is given by E c,ïâ = E c,?25 E ( ão ä )( ØO å ) æ,?25 ã å 2š E ( ão ä )( ØO å ) æ,?25e Ò,Ïâ cosθ Ïâ, sinθ Ïâ = J sinθ?25 where m y and m Ò are the mass of target and residual nucleus, respectively, and N g = P N g ( I Ø,Óè á Óè ) G ( I Ø,é@A á é@a ) where J is the Jacobian factor J = we c,?25 /E c,ïâ [64]. Examples of the experimental cross sections in CM frame for the 56 Fe(α,n) reaction with Eα = 13.5 MeV at angles of and are shown in Fig We can note that the cross sections are very similar at backward angles. Slowly changing cross sections with angle usually indicates that the compound reaction mechanism dominates. In Appendix A, the cross sections of 54 Fe(α,n), 56 Fe(α,n), 57 Fe(α,n), and 58 Fe(α,n) reactions for different emission angles at 13.5 MeV bombarding energy are shown, and the numerical values are tabulated in Tables A.1, A.2, A.3, and A.4, respectively. Q-values for (α,n) reactions are listed in Table 4.1. In addition, it is impossible to have a contribution from (α,2n) reactions with Eα=13.5 MeV to the measured cross sections due to Q-values. Table 4.1: Q-values for neutron channel. Reaction Q-value (MeV) 54 Fe + α n + 57 Ni Fe + α n + 59 Ni Fe + α n + 60 Ni Fe + α n + 61 Ni -3.58

59 59 Figure 4.1: Experimental cross sections of neutrons in CM frame measured at angles 135 (left) and 146 (right) with 13.5 MeV alpha beam from 56 Fe(α,n) reaction. The corresponding excitation energies Ex of residual nickel nuclei can be calculated from the neutron emitted energy En by the following equation E o = E æ,ïâ + Q ë,c ( ì ìer ) E c,ïâ, (4.1) where Qα,n is the Q-value of the (α,n) reaction, Eα,CM is the incident energy in CM frame (Eα,CM =(A-4/A) Eα,lab), and A is the mass number of compound nucleus. The fraction (A/A-1) is considered for the recoil effect. Hence, if the compound nucleus decays into the ground state of the residual nucleus (Ex = 0), the neutrons will be emitted with the maximum energy Enmax. The lower excited states are populated by high emission neutrons Angular distributions In direct and pre-equilibrium reactions, the outgoing particles favor the emission in the forward direction. In contrast, the particle emission in compound reactions is symmetric about 90 o in the CM frame, so the analysis of the angular distribution of evaporated particles is important to study the reaction mechanism. In our experiments, in order to obtain the angular distribution of emitted neutrons, the differential cross sections were integrated over selected energy intervals. Figure 4.2 shows the experimental angular distributions for the reactions 54 Fe(α,n) 57 Ni, 56 Fe(α,n) 59 Ni, 57 Fe(α,n) 60 Ni, and 58 Fe(α,n) 61 Ni with 13.5 MeV alpha beam integrated over different energy intervals of emitted neutrons. It is noticeable that the symmetry feature is obvious for all measured

60 angular distributions from all nuclei, which indicates that the compound reaction mechanism is dominant. 60 Figure 4.2: Experimental angular distributions of emitted neutrons from 54 Fe(α,n) 57 Ni, 56 Fe(α,n) 59 Ni (Top), and 57 Fe(α,n) 60 Ni, 58 Fe(α,n) 61 Ni (bottom) reactions with 13.5 MeV alpha beam. The cross sections have been multiplied by 4π Theoretical calculations Energy spectra of outgoing neutrons It follows from the analysis of experimental angular distributions that angular distributions are symmetric about 90 degrees in the CM frame which indicates the dominance of the compound reaction mechanism. Therefore, theoretical calculations were performed in the framework of the statistical Hauser-Feshbach (HF) theory using the HF2002 code [59] and EMPIRE code [19].

61 61 According to Hauser-Feshbach theory, both the transmission coefficients of outgoing particles and the level density of residual nuclei are needed to calculate reaction cross sections including spectra of evaporated particles. The transmission coefficients for neutrons, protons, and α-particles were calculated by using the optical model potentials, which are provided by the database of Reference Input Parameter Library (RIPL-3) [5]. The optical potential parameters (OMP) for α-particles were taken from work of Avrigeanu et al [52], and for neutrons and protons from work of Koning et al [53]. These OMP parameters are listed in Appendix B. For inputs of the nuclear level density, the following models were tested. The Fermi gas model (FGM) is employed with parameters that are determined from: (1) von Egidy systematics 2005 [44], and (2) von Egidy systematics 2009 [48], and Gilbert-Cameron model (GCM) with Arthur parameters [50]. In calculations, the level scheme of discrete levels was used up to the energy of Umax, where the levels scheme is considered to be complete. The Umax represents the excitation energy cutoff for discrete levels. The level density model was used above Umax. The values of Umax are available from the RIPL-3 database [5], and they are summarized in Table 4.2 for the residual nuclei 57 Ni, 59 Ni, 60 Ni, and 61 Ni. The values of parameters that are used from von Egidy systematics (2005) and von Egidy systematics (2009) are summarized in Table 4.3 for the reactions 54 Fe(α,n) 57 Ni, 56 Fe(α,n) 59 Ni, 57 Fe(α,n) 60 Ni, and 58 Fe(α,n) 61 Ni. In EMPIRE calculations, the Gilbert-Cameron model (GGM) [7], which consists of the constant temperature model and the Fermi-gas model is utilized with the Arthur systematics [50] for level density parameters (more details in section 2.7.5). The pairing energy is calculated as = n(12/wa), where n=0, 1, and 2 for odd-odd, odd-a, and even-even nuclei, respectively. The spin cut-off factor σ(e) is given by the Fermi-gas model σ " (E) = A "/ wa(e ) above the matching energy Em, and it is given as a linear interpolation between the discrete levels and the spin cut-off value from previous equation below Em. Table 4.4 presents the asymptotic level density parameters a (A), the shell corrections δw(z, N), the pairing energies, the nuclear temperatures T, the energy shifts Eo, and the matching energy Em values.

62 62 In Fig. 4.3, the spin cut-off parameters which are determined from von Egidy 2005, von Egidy 2009, and Arthur systematics are plotted as a function of excitation energy for the residual nuclei 57 Ni, 59 Ni, 60 Ni, and 61 Ni. Moreover, the experimental spin cut-off values σdis in the region of discrete levels were calculated. These values are obtained from known spins of low-lying discrete levels taken from the RIPL-3 database [5]. The values of the average energy E$ were calculated based on excitation energies of Nmax individual levels which are considered to constitute the complete level scheme. The values of E$ and σdis are tabulated in table 4.5, and also plotted in Fig It can be seen from Fig. 4.3 that for all four nuclei, the spin cutoff functions σ(e o ) calculated with models increases with the excitation energy Ex. At low excitation energy, the spin cut-off function (based on linear interpolation formula) from the Arthur systematics matches the experimental data from discrete levels, and at higher excitation energies, it approaches the rigid body values from von Egidy 2005 systematics. The spin cut-off values of von Egidy 2009 systematics are lower compared to the rigid body values in the all excitation energy region but higher than those from Arthur systematics at lower excitation energies. The feature that the spin cut-off curve based on GCM parametrized by Arthur systematics is not smooth at the matching energy Em is explained by the fact that the EMPIRE calculations do not use the derivative condition for the spin cutoff parameter when it transitions from the constant temperature model to the Fermi-gas model. The level density ρ(e) above the discrete level region were calculated from Fermi gas model (FGM) based on von Egidy 2005, and von Egidy 2009 systematics, and from Gilbert-Cameron model with Arthur parametrization as implemented in Empire reaction code [19]. The density of the discrete levels per MeV for residual nuclei was obtained from the RIPL-3 database by counting the number of levels up to the excitation energy Umax. Figure 4.4 shows the density of discrete levels and the model level density ρ(e) as a function of excitation energy E with different systematics for all the residual nuclei. The level density parameter a and the backshift parameter E1 for 57 Ni nucleus from Egidy 2005 and Egidy 2009 are first calculated from the given formulas and then adjusted, so that the model level densities at low energies are consistent with the density

63 63 of discrete levels. It was found that initial parameters estimated for 57 Ni with given global formulas resulted in level densities which highly overestimate the density of discrete levels at low energies. It is noticeable for all considered nuclei that at high energies the level densities from Egidy 2005 systematics are higher than those from von Egidy 2009 and Arthur systematics by factor of ~ 1.5 and 2, respectively. Table 4.2: Excitation energies Umax up to which the level scheme is considered to be complete. Element Umax (MeV) 57 Ni Ni Ni Ni 2.90 Table 4.3: Parameters for the Fermi gas (FGM) level density model from Von Egidy systematics (2005) [44] and Von Egidy systematics (2009) [48] used in our calculations. von Egidy systematics (2005) von Egidy systematics (2009) Nucleus a (MeV -1 ) E1 (MeV) a (MeV -1 ) E1 (MeV) Pa` (MeV) 57 Ni Ni Ni Ni

64 Table 4.4: Gilbert-Cameron level density parameters from Arthur systematics [50] used in our calculations. 64 Nucleus a (A) (MeV -1 ) δw(z,n) (MeV) (MeV) T (MeV) Eo (MeV) Em (MeV) 57 Ni Ni Ni Ni Table 4.5: Parameters extracted from discrete levels (spin cut-off parameter σdis at the average energy E$) [5]. Nucleus Eî (MeV) σ dis 57 Ni 2.8± Ni 2.6± Ni 3.3± Ni 1.9±

65 65 Figure 4.3: Spin cut-off parameter as function of excitation energy of 57 Ni, 59 Ni (top), and of 60 Ni, 61 Ni (bottom) nuclei. The black point is the experimental spin cut-off derived from discrete levels.

66 66 Figure 4.4: Level densities of 57,59 Ni (top) and 60,61 Ni (bottom). Solid lines are calculations with FGM (Egidy2005 sys & Egidy2009 sys) and GCM (Arthur sys). Histogram is the density of discrete levels.

67 4.2.2 Angular distributions The theoretical calculations for the angular distribution of (α,n) reactions are performed by using the Hauser-Feshbach code (HF2002) developed at Ohio University [59]. Input level density models and spin cutoff parameterization were based on von Egidy 2005 and von Egidy 2009 and Arthur systematics described above. As an example, Fig. 4.5 shows the angular distributions calculated with Egidy 2005 and von Egidy 2009 models for the reaction 56 Fi(α,n) in the neutron energy intervals MeV and MeV. These neutron energy intervals correspond to excitation energy intervals of MeV and MeV in 59 Ni respectively. It can be noticed that there is a difference between the two curves in the corresponding excitation energy interval MeV while in the energy interval of discrete levels MeV, the difference is negligible. The calculated angular distributions in the discrete range are expected to be very similar because we use the same spin and parities for discrete levels in calculations. The small difference is explained by competition in orbital momentum space so that any changes of spin cutoff values in continuum region would affect the angular distribution of discrete levels. In the excitation energy range between 2.4 MeV and 4.0 MeV, the values for the spin cut-off parameter for 59 " Ni are calculated to be σ IóQ ô" = 11.7 and " σ IóQ ô" ¾ =8.6 at the average excitation energy of around 3 MeV. As it will be discussed in the following section, the angular anisotropy can be determined as a ratio of cross sections at 0 o and 90 o, and it is equal 1.16 and 1.30 for Egidy 2005 and Egidy 2009 systematics, respectively. Hence, the anisotropy is expected to be larger for smaller value of the spin cutoff parameter and this is in agreement with the measurements from Ref. [65]. 67

68 68 Figure 4.5: Angular distributions based on Egidy 2005 and Egidy 2009 systematics from 56 Fe(α,n) reaction in two different excitation energy intervals. The cross sections have been multiplied by 4π to convert the units from (mb/sr) to (mb).

69 4.3 Comparison between experimental results and theoretical calculations The main focus will be on comparison of experimental angular distributions with that calculated using different level density models which imply different spin cutoff parameter function. As it has been discussed earlier the anisotropy of the symmetric angular distribution is sensitive to the spin cutoff parameter determining the spin distribution for levels populated by neutrons from (a,n) reaction. To access the degree of consistency between experimental and calculated angular distribution, the criteria based on χ " statistics has been employed in the following form } ( gõö (á Ý )/ á Ý T M. g [@é[ (á Ý )/ á Ý ) N χ " = Q^P, (4.2) (x g õö (á Ý )/ á Ý ) N where the numerator represents the difference between experimental and calculated differential cross sections at angle θ Q, the denominator represents the uncertainty of experimental points, N is the number of experimental points, k is the coefficient adjusted to minimize the value of χ ". The later coefficient is used to eliminate the difference caused by common scaling factor, so the only difference in shape would be estimated. The above formula implies that experimental uncertainties in denominator are estimated correctly, but that is not always the case. In this work uncertainty of experimental points is based on counting statistics only. Therefore, we assume that possible deviations of experimental points caused by any extra systematic uncertainties might increase values χ " calculated according the above formula and using the criteria based on χ " would not be possible. In this case the following ratio is suggested to be calculated 69 R = N N, (4.3) Aø " where χ 5ù is the value of χ " from the best fit which is obtained with above formula where dσ`2?`(θ Q )/dθ Q is replaced by the polynom f nop (x) = a + bx " + cx. (4.4) Here x = cos(θ), and the coefficients a, b, and c of which are determined by fitting it to experimental data points dσ nop (θ Q )/dθ Q. One can assume that the value of R would represent the relative difference of actual χ " value from its average determined by χ "

70 distribution with N-1 degrees of freedom. Such a deviation determines probability of rejecting or accepting the angular distribution model dσ`2?`(θ Q )/dθ Q. The obtained values of probability (P-values) are tabulated in the discussion part (Sec. 4.4) for all reactions. The other method of analysis is related to analysis of the anisotropy of an angular distribution. As a measure of anisotropy the ratio of R 2côû = σ(θ = 0 )/σ(θ = 90 ) was used. The experimental angular distributions were fitted with a polynom f nop (x), and the anisotropy ratio Ranys,exp was calculated. Uncertainty based on errors of fitting coefficients was also determined. For HF model calculations, the ratio Ranys is directly obtained from calculated angular distributions. In addition, the asymmetry ratio Rasym is calculated for the experimental angular } distributions by using R 2ûô = ( ø } σ Q^P Q,ùýÊÇ2Ê û /N ù )/( A Q^P σ Q,52`MÇ2Ê û 70 /N 5 ), where Nf and Nb are number of points at forwards and backwards angles. The Rasym should be around one within uncertainties for symmetric angular distributions. In our case, if this value differ significantly from one, such angular distributions were not used in an analysis. Rasym values are presented for each reaction in the discussion section 4.4. Angular distributions have been integrated over selected energy intervals of about 2 MeV each. Energy intervals were increased incrementally from low energy neutrons to highest neutron energies for a particular neutron spectrum. Those angular distributions which were asymmetric (Rasym value is greater or less than one), have been excluded from further analysis. For some angular distributions, if individual points were found to deviate markedly from a general trend, such points were also excluded from an analysis Fe(α,n) reaction For 54 Fe(α,n) reaction, the experimental cross sections (at ) and theoretical cross sections calculated with different models are shown in Fig The known energy, spin and parity of discrete levels below the established excitation energy of Umax = 4.61 MeV (see section 4.2.1) are used to calculate the cross sections for neutrons with energies above 2.3 MeV. It can be seen in Fig. 4.6 that the theoretical cross sections produce the discrete levels structure in this energy region. It is also observed that in the continuum

71 71 region (at low neutron energies), calculated cross sections using Egidy 2005 parameters are larger than that calculated with Egidy 2009 and Arthur parameters by factor of about 1.5. In the discrete energy region (at higher neutron energies), the cross sections from Egidy 2005 and Egidy 2009 systematics underestimate the experimental cross sections while calculations with Arthur systematics agree with experimental data points well. One therefore concludes that the level density model based on Gilbert and Cameron approach with Arthur parameter systematics reproduces experimental data points better compared to Fermi-gas model with Egidy systematics. For this nucleus we also found a small disagreement in position of high energy discrete peaks in neutron spectra. This issue is currently addressed by code developers. Both experimental and calculated neutron angular distributions from the reaction 54 Fe(α,n) for selected neutron energy intervals are displayed in Fig Calculations were performed with von Egidy 2005, von Egidy 2009, and Arthur parameter systematics. The absolute normalization of theoretical angular distributions has been performed to match the experimental data points. The χ 2 and R values are calculated from Eqn. (4.2) and Eqn. (4.3) for each energy interval, and Table 4.6 and Table 4.7 show these values, respectively. The best fit values included in tables calculated by fitting the experimental data points with the polynomial function f(x) from Eqn. (4.4) as mentioned in Sec 4.3.

72 72 Figure 4.6: Energy spectrum from 54 Fe(α,n) reaction. The points are the experimental data and the solid lines indicate the HF calculations with Egidy 2005 and Egidy 2009 models and Empire calculations with Arthur model.

73 73 Figure 4.7: Neutron angular distributions from 54 Fe(α,n) reaction for different energy intervals. The points are the experimental data and the solid lines are calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π.

74 74 Table 4.6: The χ 2 values from neutron angular distributions of 54 Fe(α,n) reaction. En is the energy interval for emitted neutrons, and Ex is the corresponding excitation energy interval for the residual 57 Ni nucleus. First two intervals were excluded from analysis because of strongly asymmetric angular distributions. En (MeV) Ex (MeV) Egidy Egidy Arthur Best fit Table 4.7: R values from angular distributions of 54 Fe(α,n) for different energy intervals. En is the energy interval for emitted neutrons, and Ex is the corresponding excitation energy interval for the residual 57 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur

75 Fe(α,n) reaction The experimental differential cross sections (dσ/de) n2û (at backward angle 146 o ) along with theoretical calculations (dσ/de)`2?` from HF and EMPIRE codes are shown in Fig For theoretical calculations, the Fermi gas model (FGM) were utilized with the parameters taken from von Egidy 2005, and von Egidy 2009 systematics, and the Gilbert-Cameron model (GCM) was based on Arthur parametrizations. Below the excitation energy of 2.71 MeV for the 59 Ni residual nucleus, the data from discrete level scheme were used in calculations. The structure produced by discrete levels is seen in the neutron spectrum above En ~ 4.9 MeV (Fig. 4.8). It is observed that theoretical neutrons cross-sections with the GCM Arthur systematics and FGM with Egidy 2005 and Egidy 2009 systematics have approximately the same slope in the continuum part of the spectrum. Absolute cross sections in continuum from Arthur systematics are in reasonable agreement with experimental data while data are overestimated by Egidy 2005 and Egidy 2009 systematics. In the discrete energy interval (En ~ MeV) Arthur calculations slightly overestimate data points while Egidy ones are in a good agreement with them. Figure 4.9 shows experimental and theoretical angular distributions for neutrons from the 56 Fe(α,n) reaction. Calculations have been performed with HF code using von Egidy 2005, von Egidy 2009, and Arthur systematics of level density and spin cutoff parameters. Calculations exhibit a symmetric angular distribution at 90 o. The calculated values for χ 2 and R for von Egidy 2005, von Egidy 2009, Arthur systematics, and for the best fit polynomial function f(x) (Eqn. 4.4) are summarized in Table 4.8 and Table 4.9.

76 76 Figure 4.8: Neutron energy spectrum from 56 Fe(α,n) reaction. The points are experimental data and the solid lines indicate the HF theoretical calculations using models.

77 77 Figure 4.9: Neutron angular distributions from 56 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π.

78 78 Table 4.8: The χ 2 values for 56 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual nucleus 59 Ni. En (MeV) Ex (MeV) Egidy Egidy Arthur Best fit Table 4.9: R values for angular distributions from 56 Fe(α,n) reaction for different energy intervals. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual nucleus 59 Ni. En (MeV) Ex (MeV) Egidy Egidy Arthur

79 Fe(α,n) reaction The measured and calculated cross sections (neutron spectra) of the 57 Fe(α,n) reaction are shown in Fig For theoretical calculations, the Fermi gas model with parameters from von Egidy 2005 and von Egidy 2009, and GCM model with Arthur parametrization were utilized. The discrete levels were used in cross section calculations below the excitation energy of 4.61 MeV, so corresponding structure can be seen in Fig above neutron energies of about 6 MeV. Moreover, the cross sections calculated with Arthur systematics seems to be in better agreement with the experimental cross sections in the continuum region, but it overestimates the data points in discrete levels region. The observed inconsistency for some particular model to describe data in both continuum and discrete level region is due to the fact that the absolute level density values predicted by level density model deviates from correct numbers. Theoretical angular distributions of evaporated neutrons for selected energy intervals from the reaction 57 Fe(α,n) are compared with experimental angular distributions in Fig The χ 2 values are calculated from Eqn. (4.2) and tabulated in Table The values of R for different energy intervals are also summarized in Table 4.11.

80 80 Figure 4.10: Neutron energy spectrum from 57 Fe(α,n) reaction. The points are the experimental data and the solid lines indicate theoretical calculations using different level density models.

81 81 Figure 4.11: Neutron angular distributions from the 57 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are HF calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections were multiplied by 4π.

82 82 Table 4.10: The χ 2 values of the 57 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 60 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur Best fit Table 4.11: R values for angular distributions from 57 Fe(α,n) for different energy intervals. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 60 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur

83 Fe(α,n) reaction The experimental cross sections at 146 o along with theoretical calculations are shown in Fig The Fermi gas model (FGM) was used in calculations with parameters taken from von Egidy 2005, and von Egidy 2009 systematics. The Gilbert- Cameron model used Arthur parameter systematics from EMPIRE code. Below the excitation energy of 2.90 MeV, the energy, parity, and spin of discrete levels were used for neutron cross-section calculations. As shown in Fig. 4.12, the structure due to discrete levels is produced in the region above En 5.5 MeV. It is observed in the region below this energy that the slope of cross sections predicted by GCM (Arthur sys) and FGM (Egidy 2005 and 2009) models is roughly the same, but absolute values exceed experimental data points. In the region of discrete levels (En greater than about 5.5 MeV), the calculations with Egidy 2005 and Egidy 2009 models describe observations better. As we explained for other nuclei, this is because of the behavior of model level densities at low and high excitation energies as displayed in Fig The experimental and theoretical angular distributions for the 58 Fe(α,n) reaction are displayed in Fig Calculations were performed with von Egidy 2005, von Egidy 2009, and Arthur systematics. The absolute scaling of each distribution has been adjusted to minimize χ 2 values. The χ 2 values are summarized in Table 4.12 and R values in Table 4.13.

84 84 Figure 4.12: Neutron energy spectrum from 58 Fe(α,n) reaction. The points are the experimental data and the solid lines indicate the HF calculations using FGM and GCM models with different systematics.

85 85 Figure 4.13: Neutron angular distributions from 58 Fe(α,n) reaction for different energy intervals. The points are experimental data and the solid lines are HF calculations with von Egidy 2005 (top), von Egidy 2009 (middle), and Arthur (bottom) systematics. The cross sections have been multiplied by 4π.

86 86 Table 4.12: The χ 2 values of 58 Fe(α,n) reaction form neutron angular distributions. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 61 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur Best fit Table 4.13: R values for angular distributions from 58 Fe(α,n) reaction for different energy intervals. En is the energy of emitted neutrons, and Ex is the corresponding excitation energy of the residual 61 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur

87 Results and discussion In this section the spin cutoff parameter from three level density models used in section 4.3 will be analyzed using our calculated and experimental data on angular distributions presented in previous section 4.3. Two types of approaches are implemented. The first one is based on using χ " values, and the second approach uses the anisotropy indicator R (Ranys and Rasym) for angular distributions. R values are presented in Tables 4.7, 4.9, 4.11, 4.13 in section 4.3, and Ranys, Rasym in Tables 4.18, 4.19, 4.20, and 4.21 in section for all measured reactions Analysis based on χ 2 If our experimental data points were well described by a theoretical model, the " calculated χ Æ values (Eqn. 4.2) would be distributed around ν = N 1 degrees of freedom (N is the number of experimental data points). There is a different distribution for each degree of freedom, and it becomes more symmetric for larger values of ν. The probability of accepting the theoretical model (P-value) is the area that lies to the right of " χ Æ value as displayed in Fig These values of probability are also available from " tables for each degree of freedom and χ Æ value. In our case, we tested the goodness of fitting of our theoretical models for angular distributions based on the normalized R value (Eqn. 4.3) (R value per ν degree of freedom) in each energy interval of outgoing neutrons. It is expected that the value of R is close to 1 for the best fit, and it has a probability of 0.5 to be larger than one. P values corresponding to experimental values of R will be presented in this section. These P values would indicate probability of acceptance of a hypothesis which is in our case, a certain model used to calculate angular distributions.

88 88 Figure 4.14: Probability versus chi-square for different values of degrees of freedom. The figure is taken from Ref. [66]. For the 54 Fe(α,n) reaction, the values of probability P are obtained and tabulated in Table In the energy intervals and MeV, probability to accept all models is high, so it is not possible to give a preference to any of these models. For lower neutron energy intervals, the angular distributions were found to be forward peaked. This might be caused by different reasons including physics ones as well as possible deficiencies related to processing of experimental data. Specific reason needs to be found out from repeated analysis of data or checked with more measurements. At this point the conclusion based on obtained P values for these energy intervals, that the Arthur systematics works best, need to be taken with caution. The probability values for the angular distributions from 56 Fe(α,n) reaction are summarized in Table Based on these numbers one can definitely conclude that the spin cutoff parameter model based on Egidy2005 systematics is ruled out. It is more

89 89 difficult to choose between Egidy 2009 and Arthur models. One might give a slight preference to the Arthur model, however, most likely that no one of these models are perfectly correct. The real spin cutoff parameter is closer to the Egidy 2009 value at around 4.5 MeV of excitation energy ( MeV neutron energy interval), and it is closer to Arthur value at lower excitation energies. For the 57 Fe(α,n) reaction, the P values strongly support Egidy 2005 model in all energy intervals (Table 4.16). Egidy 2009 and Arthur models fail completely to reproduce angular distributions in the and MeV energy intervals. Finally, the obtained probabilities for 58 Fe(α,n) reaction are listed in table The FGM (Egidy 2005 sys) shows a very low probability to describe experimental angular distributions in all energy intervals, so this model can be completely rule out. Two other models which are Egidy 2009 and Arthur ones show the same degrees of probability for and intervals, but Arthur model has a slight preference in energy interval. Again, one may conclude that the real spin cutoff model function is most likely different from all models used here for analysis. However, from three models tested here, the Arthur model should be given the preference. Angular distributions for and energy intervals for the 58 Fe(α,n) reaction were not used because of similar problems (asymmetry of obtained angular distributions) with experimental data points as discussed above for the 54 Fe(α,n) reaction.

90 90 Table 4.14: P-values for 54 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 57 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy Arthur Table 4.15: P-values for 56 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 59 Ni nucleus. En (MeV) Ex (MeV) Egidy e < 1e-4 1e-4 Egidy < 1e-4 1e-3 Arthur e-3

91 91 Table 4.16: P-values for 57 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 60 Ni nucleus. En (MeV) Ex (MeV) Egidy Egidy e Arthur Table 4.17: P-values for 58 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy for the residual 61 Ni nucleus. First two intervals were excluded because of experimental angular distributions are asymmetric. En (MeV) Ex (MeV) Egidy e-4 <1e-4 2e-4 Egidy e Arthur

92 Analysis based on anisotropy of angular distributions Another approach to test our theoretical models is analyzing the anisotropy of the angular distributions. The anisotropy Ranys was determined as a ratio of cross sections at zero and 90 degrees angles for experimental as well as for theoretical angular distributions as described in Sec Tables 4.18, 4.19, 4.20, and 4.21 summarize the Ranys values for the reactions 54 Fe(α,n), 56 Fe(α,n), 57 Fe(α,n), and 58 Fe(α,n). These values are also plotted as a function of the excitation energy in Fig. 4.15, 4.16, 4.17, and For the 54 Fe(α,n) reaction, the high value of asymmetry ratio Rasym (see section 4.3 for definition of Rasym) indicates that experimental angular distribution is not symmetric in the excitation energy interval MeV, it is also shown in Fig. 4.7 that these points are forward peaked, so this angular distribution was excluded from the analysis and the value of Ranys, exp at Ex= 4.8 MeV is excluded from Fig For this reaction the discrete levels with specific spins and parities were used in calculations up to 2.2 MeV of excitation energy (average energy is 1.1 MeV) which corresponds to the first point in Figure The second point (at average energy of 3.3 MeV) represents calculations performed with model level densities including corresponding model spin cutoff parameters. We can say from Fig that the calculations based on all models can be utilized to represent both data points. This is expected because spin cutoff parameters for the residual 57 Ni nucleus from all models differ insignificantly in the excitation energy region up to 3.3 MeV. For the 56 Fe(α,n) reaction, it can be observed from Fig that GCM based on Arthur parameters is best to describe the experimental angular distribution for the reaction 56 Fe(α,n) at low and high excitation energies. This strongly supports the utilized spin cutoff model in the calculations which is a linear interpolation between the spin cutoff values from discrete levels and the spin cutoff value from the equation σ " (E ) = A "/ wa(e ) at the matching energy Em of the GCM model. FGM models with both von Egidy parameter systematics are consistent with experimental data points at higher excitation energy but definitely fails at low excitation energy. This indicates that the spin cutoff parameter is consistent with rigid model values at higher excitation energies. At low excitation energies it tends to be lower compared to rigid body

93 93 estimates. For the 57 Fe(α,n) reaction, we show the Ranys as a function of excitation energies in Fig At high excitation energy, it is seen that the data points are well reproduced by FGM with rigid body model of spin cutoff parameter (Egidy 2005 model) while at low excitations (including discrete levels region), Egidy 2009 parameters are better for representing experimental angular distributions. GCM model with Arthur systematics seems to overestimate the anisotropy indicating underestimating spin cutoff parameter values in the whole excitation energy region. The observed tendency is consistent with that observed for the 56 Fe(α,n) reaction discussed above. The spin cutoff parameters are consistent with rigid body estimates of Egidy 2005 model at higher excitation energies and tends to be lower than that at low excitation energy region. The results of anisotropy ratio Ranys from the reaction 58 Fe(α,n) (Fig. 4.18) show the Arthur parameterization with a linear interpolation of the spin cutoff parameter at low excitation energies to be best compared to other models. This again supports reduction of the spin cutoff parameter compared to rigid body estimates at low excitation energies.

94 94 Table 4.18: Ranys for 54 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy. En (MeV) Ex (MeV) Ex,center (MeV) Rasym, exp 1.36 (7) 1.00 (6) 1.33 (20) Ranys, exp 1.69 (21) 1.07 (13) 2.20 (32) Ranys, Egidy Ranys, Egidy Ranys, Arthur Figure 4.15: Anisotropy ratio Ranys as a function of excitation energy from the reaction 54 Fe(α,n).

95 95 Table 4.19: Ranys for 56 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy. En (MeV) Ex (MeV) Ex,center (MeV) Rasym, exp 1.07 (2) 0.98 (2) 1.02 (3) 1.02 (5) Ranys, exp 1.20 (4) 1.18 (8) 1.47 (4) 2.17 (20) Ranys, Egidy Ranys, Egidy Ranys, Arthur Figure 4.16: Anisotropy ratio Ranys as a function of excitation energy from the reaction 56 Fe(α,n).

96 96 Table 4.20: Ranys for 57 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy. En (MeV) Ex (MeV) Ex,center (MeV) Rasym, exp 0.98 (2) 1.03 (3) 1.08 (5) 1.08 (10) 1.14 (30) Ranys, exp 1.08 (8) 1.22 (4) 1.23 (8) 1.70 (15) 2.15 (30) Ranys, Egidy Ranys, Egidy Ranys, Arthur Figure 4.17: Anisotropy ratio Ranys as a function of excitation energy from the reaction 57 Fe(α,n).

97 97 Table 4.21: Ranys for 58 Fe(α,n) reaction, where En is the energy of emitted neutrons, and Ex is the corresponding excitation energy. En (MeV) Ex (MeV) Ex,center (MeV) Rasym, exp 0.89 (2) 0.98 (4) 1.06 (7) 1.26 (14) Ranys, exp 1.21 (3) 1.22 (3) 1.49 (4) 1.89 (15) Ranys, Egidy Ranys, Egidy Ranys, Arthur Figure 4.18: Anisotropy ratio Ranys as a function of excitation energy from the reaction 58 Fe(α,n).

98 98 5 SUMMARY AND CONCLUSIONS The purpose of this work was to study the spin distribution of 57,59,60,61 Ni nuclei at excitation energies higher than discrete levels known from established level scheme in literature. The method used was the experimental measurements and analysis of the angular distribution of neutrons from 54,56,57,58 Fe(α,n) 57,59,60,61 Ni reactions. This method was based on the early finding [8] that the magnitude of the anisotropy is determined by the spin cutoff parameter responsible for spin distributions in the Fermi-gas level density model. Neutrons have been measured from 54,56,57,58 Fe(α,n) reactions with time of flight technique by using alpha beam of 13.5 MeV energy. The experiment was carried out at the Edwards Accelerator Laboratory with the swinger time of flight neutron facility. The emitted neutrons were detected with a NE213 neutron detector set up inside the wellshielded underground TOF tunnel. In order to study the angular distributions, the outgoing neutrons were measured at different angles. The same procedures were followed for time and energy calibration for all reactions as described in section The experimental differential cross section has been obtained at different angles. The experimental cross sections and angular distributions have been compared with theoretical calculations using HF reaction code [59] and EMPIRE code [19]. Different inputs of level density models using different prescriptions for the spin cutoff parameter have been tested. Those are from Fermi gas model (FGM) with von Egidy 2005 [44], and von Egidy 2009 [48] systematics, and Gilbert- Cameron model (GCM) with Arthur systematics [50]. The analysis of angular distributions performed with criteria based on both χ 2 statistical distribution and on analysis of the magnitude of the anisotropy (0/90 degrees cross section ratio) resulted in the following conclusions. First of all, the analysis revealed that some experimental data have deficiencies related to, most likely, inaccurate estimates of errors of experimental points and/or improper procedure used for background measurements and its subtraction. This resulted, in some cases, in asymmetric angular distributions, which is not expected from these kinds of measurements. Some of the experimental points deviated significantly beyond

99 99 statistically acceptable range. All of these deficient measurements were excluded from analysis. It is strongly advised that these data should be revised either by repeating analysis of existing data or by repeating actual measurements with better background conditions. Other measurements which were considered to be normal have been used to deduce physics conclusions about the spin cut-off parameter models. It has been found that for the 57 Ni nucleus (produced from 54 Fe(α,n)), it is not possible to distinguish between tested models for the spin cutoff parameters. For the 59 Ni nucleus (produced from 56 Fe(α,n)), the Egidy 2005 model can be completely ruled out with high probability (more than 95%). The preference should be given to Arthur model [19] which used linear interpolation of the spin cut-off parameter between discrete levels and the Fermi-gas model at the matching point in the Gilbert and Cameron approach. For the 60 Ni nucleus (produced from 57 Fe(α,n)), the Egidy 2005 model is best at higher excitation energies (5-9 MeV), but Egidy 2009 model implying lower values of the spin cut-off parameter is better below 4 MeV of excitation energies. For the 60 Ni nucleus (produced from 58 Fe(α,n)), the Egidy 2005 can be completely ruled out. The Arthur model generally better describes experimental data points compared to Egidy 2009, although, the Egidy cannot be completely ruled out. The general conclusion is that at higher excitation energy, the model of the spin cutoff parameter based on rigid body model is better able to describe data. These were used in Egidy 2005 model as well as in the Arthur model at high excitation energy. The only difference between them is they use different parameterization; Egidy generally gives higher values compared to Arthur model. At lower excitation energies, data show that spin cutoff parameter needs to be smaller compared to rigid body estimates. It can be achieved either by using Arthur parametrization implying the linear interpolation to much smaller values derived from discrete level scheme or by using the spin cutoff parameter formula Eqn (2.39) from Egidy 2009 systematics [48], which also gives smaller values of spin cutoff parameters. The Arthur linear interpolation methods appears to be more reliable for global approach although specific parameterization is likely not very precise, as our data show. The reduction of the spin cut-off parameter at low excitation energies has been explained in Ref [67] due to the strong pairing correlation which considers the

100 nucleons as condensed BCS pairs, and it results in the moment of inertia I(E o ) to be less than the rigid body value I ħ " = A /. 5.1 Outlook 100 In this work, the symmetric experimental angular distributions of outgoing neutrons from alpha induced reactions on iron isotopes showed that the compound reaction mechanism is dominant. However, we found possible problems with data since some of experimental data points deviated significantly from general trend of distribution curves, so it is important to perform this experiment with better background conditions by reducing the background neutrons from beam collimators and the beam dump. It has been shown in this dissertation that measuring angular distribution of neutrons from alpha induced reactions allows one to study the energy dependence of the spin cutoff parameter. Therefore, studying more elements which have several stable isotopes such as 58,60,61,62,64 Ni, 46,47,48,49,50 Ti, and 50,52,53,54 Cr would be helpful to make general conclusions on systematic behavior of the spin cutoff parameter versus mass, excitation energy and, more importantly, isospin value (N-Z)/A. Our measurements also indicate that the reduction of the spin cutoff parameter due to, presumably, pairing correlation effect should be taken into account in phenomenological models currently used in calculations of the nuclear level density.

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105 105 APPENDIX A: EXPERIMENTAL DIFFERENTIAL CROSS SECTIONS A.1 Cross sections of 54 Fe(α,n) reaction

106 106

107 107

108 108

109 109 Figure A.1: Neutron differential cross sections of 54 Fe(α,n) reaction for Eα = 13.5 MeV at different angles.

110 A.2 Cross sections of 56 Fe(α,n) reaction 110

111 111

112 112

113 113

114 114 Figure A.2: Neutron differential cross sections of 56 Fe(α,n) reaction for Eα = 13.5 MeV at different angles.

115 A.3 Cross sections of 57 Fe(α,n) reaction 115

116 116

117 117

118 118 Figure A.3: Neutron differential cross sections of 57 Fe(α,n) reaction for Eα = 13.5 MeV at different angles.

119 A.4 Cross sections of 58 Fe(α,n) reaction 119

120 120

121 121

122 122

123 123 Figure A.4: Neutron differential cross sections of 58 Fe(α,n) reaction for Eα = 13.5 MeV at different angles.

124 Table A.1: Numerical values for the experimental cross sections for outgoing neutrons from the 54 Fe(α,n) reaction at Eα = 13.5 MeV. No data are in the blank cells. 54 Fe 20 deg 34 deg 45 deg 62 deg 76 deg 90 deg 118 deg 135 deg 146 deg dσ/de ECM (mb/ Error (MeV) MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) dσ/de (mb/ Error MeV) 124 dσ/de (mb/ Error MeV)

125 125 Table A.1: continued

126 Table A.2: Numerical values of the experimental cross sections for outgoing neutrons from the 56 Fe(α,n) reaction at Eα = 13.5 MeV. No data are in the blank cells. 56 Fe 20 deg 34 deg 45 deg 62 deg 76 deg 90 deg 118 deg 135 deg dσ/de ECM (mb/m Error (MeV) ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev) dσ/de (mb/m Error ev)

127 127 Table A.2: continued

128 128 Table A.3: Numerical values of the experimental cross sections for outgoing neutrons from the 57 Fe(α,n) reaction at Eα = 13.5 MeV. No data are in the blank cells. 57 Fe 35 deg 53 deg 72 deg 90 deg 109 deg 128 deg 146 deg ECM (MeV) dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error dσ/de (mb/m ev) Error

129 129 Table A.3: continued

130 130 Table A.4: Numerical values of the experimental cross sections for outgoing neutrons from the 58 Fe(α,n) reaction at Eα = 13.5 MeV. No data are in the blank cells. 58 Fe 20 deg 34 deg 45 deg 62 deg 76 deg 90 deg 118 deg 135 deg dσ/de ECM (mb/m Error (MeV) ev) dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error dσ/de (mb/ MeV) Error

131 131 Table A.4: continued

132 132 APPENDIX B: OPTICAL MODEL PARAMETERS Table B.1: Optical model parameters for α-particles on target nuclei 54 Fe, 56 Fe, 57 Fe, and 58 Fe at Eα = 13.5 MeV. The potential depths are given in MeV and the radial parameters in fm [52]. VV WV WS VSO WSO r ÆÅ a ÆÅ r ÇÅ a ÇÅ r Ç» a Ç» r Æ»Ë a Æ»Ë r Ç»Ë a Ç»Ë r Ï Table B.2: Optical model parameters for neutrons in exit channels n+ 57 Ni, n+ 59 Ni, n+ 60 Ni, and n+ 61 Ni. The potential depths are given in MeV and the radial parameters in fm [53]. VV WV WS VSO WSO r ÆÅ a ÆÅ r ÇÅ a ÇÅ r Ç» a Ç» r Æ»Ë a Æ»Ë r Ç»Ë a Ç»Ë r Ï

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