Bc. Ingrid Knapová. Photon strength functions in 168 Er from multi-step gamma cascade measurement at DANCE

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1 MASTER THESIS Bc. Ingrid Knapová Photon strength functions in 168 Er from multi-step gamma cascade measurement at DANCE Institute of Particle and Nuclear Physics Supervisor of the master thesis: Study programme: Study branch: doc. Mgr. Milan Krtička, Ph.D. Physics Nuclear and Particle Physics Prague 217

2 I declare that I carried out this master thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 6 subsection 1 of the Copyright Act. In Prague date signature of the author i

3 Title: Photon strength functions in 168 Er from multi-step gamma cascade measurement at DANCE Author: Bc. Ingrid Knapová Institute: Institute of Particle and Nuclear Physics Supervisor: doc. Mgr. Milan Krtička, Ph.D., Institute of Particle and Nuclear Physics Abstract: Subject of the thesis is study of photon strength functions describing the gamma decay of the nucleus. During last 5 years a number of theoretical models for these quantities were proposed, however, their accuracy is still debatable and its verification has recently been a subject of intensive experimental and theoretical research. In this work measurement of multi-step gamma cascades following the radiative neutron capture on target nucleus 167 Er is used to analyse photon strength functions. The experiment was performed with DANCE calorimeter located in Los Alamos Neutron Scattering Center. The experimental spectra are compared with Monte Carlo simulations of electromagnetic decay using the DICEBOX algorithm, based on assumptions of Extreme Statistical Model. Comparison of experimental and simulated spectra is a fundamental tool for studying correctness of theoretical models of photon strength functions. This study provides information mainly about E1 and M1 photon strength functions, especially properties of the so-called scissor mode are analysed. Keywords: DANCE, Level density, Photon strength functions, Scissors mode, Extreme statistical model ii

4 I would like to deeply thank to doc. Mgr. Milan Krtička for his help, countless discussions, willingness, and most of all patience throughout the entire process of supervising this thesis. I also wish to thank to my family for their support and encouragement. iii

5 Contents Introduction 3 1 Level density and photon strength functions Level density Photon strength functions Fluctuation properties of partial widths E1 photon strength function M1 photon strength function E2 photon strength function Experiment and data processing DANCE detector array Data acquisition Detector granularity Time alignment and energy calibration Experimental sum-energy and MSC spectra Influence of parameters on MSC spectra Coincidence window Detection threshold Experimental background Time-independent background Time-dependent background Scattered neutrons Neutron capture on target impurities Modelling Extreme Statistical Model The modelling algorithm The detector response Results and discussion Model combinations of LD and PSF Comparison of experimental and simulated MSC spectra Level density E1 photon strength function M1 photon strength function Sum-energy spectra and population of 194 kev isomeric state Conclusions 52 Bibliography 53 List of Figures 56 List of Tables 59 1

6 List of Abbreviations 6 2

7 Introduction Aim of this thesis is to study γ-decay of excited nuclei following non-relativistic nuclear reactions. The nuclear hamiltonian is incredibly complicated, therefore it is not possible to formulate fundamental equations describing the process of γ-ray emission. This problem can be solved by introducing simplified models for description of the nucleus such as photon strength functions, also called γ-ray or radiative strength functions. Properties of nuclear levels and transitions between them are in medium heavy and heavy nuclei usually known only at low excitation energies. In this region individual levels can be observed due to large spacing between them. The level density increases with excitation energy and it becomes difficult to obtain information on transitions between these high-lying levels forming the so-called quasicontinuum. It is believed that the nucleus in quasicontinuum can be described by nuclear levels density and photon strength functions. By using these statistical quantities one is able to predict smooth, average properties of the γ-ray spectrum. Knowledge of these quantities is useful in the field of nuclear reactors and nuclear astrophysics. Since the introduction of photon strength functions in early 5 s, a large amount of experimental data on this topic has been obtained. The study of photon strength functions was started mainly by photonuclear experiments and slow-neutron capture reactions. Later also experiments with elastic and inelastic scattering of photons, electrons, protons and neutrons at various energies were carried out. Besides that, light- and heavy-induced fusion reactions provided some information. In this thesis photon strength functions are studied by measuring γ-ray emission induced by radiative neutron capture reaction on the target nucleus 167 Er. Capturing states above neutron separation energy have very narrow widths.1 ev, which corresponds to lifetimes of the states 1 14 s. Neutron capture is an example of reaction going via compound nucleus and according to Bohr and his concept [1], long lifetimes of the capturing states are caused by sharing the projectile energy among all other constituents of the nucleus. Within this theory the decay mode of the compound nucleus is independent of its origin. A scheme of neutron capture reaction in a heavy nucleus is shown in Fig. 1. A neutron is captured in the nucleus with mass number A, the excited compound nucleus with mass number A+1 and energy given by the sum of the neutron separation energy S n and the neutron kinetic energy E n is formed. A typical neutron separation energy in the region of rare-earth nuclei is 6-9 MeV, in the case of 168 Er it is MeV. The resulting A+1 nucleus is a well-defined quantum mechanical state called neutron resonance and it decays via emission of primary γ-ray and emission of secondary γ-rays continues until the nucleus reaches the ground state. The decay of a medium-heavy or heavy nucleus goes via large number of levels and the γ-ray spectrum is therefore very complex. The spectrum is dominated by lines corresponding to high-energy primary transitions and those belonging to the secondary transitions between low-lying levels. In addition to these strong transitions, there are weak and very weak lines corresponding to low-energy primary 3

8 Figure 1: A scheme of the (n, γ) reaction [2]. transitions and secondary transitions between levels with medium or high excitation energies. These weak lines form a continuous component of the spectrum and this hidden part provides an important amount of information on photon strength functions. We investigated γ-ray spectra from measurement at DANCE - Detector for Advanced Neutron Capture Experiments - detector arranged in Los Alamos Neutron Scattering Center (LANSCE) in New Mexico, USA. The target nucleus 167 Er was irradiated by neutrons from spallation source and experimental γ-ray spectra were compared with simulated spectra from Monte Carlo modelling based on postulates of Extreme Statistical Model and employing various assumptions on the level density and photon strength functions. In Chap. 1 theoretical and experimental knowledge on nuclear level density and photon strength functions is summarized, all models tested in our simulations are described. Chap. 2 is dedicated to experimental setup and analysis of the experimental spectra. Chap. 3 contains description of the DICEBOX algorithm used for modelling of the γ-decay and the Geant4 simulation of the detector response. Comparison between experimental and simulated spectra is present in Chap. 4. Chap. 5 includes conclusion and possible future outlook. 4

9 1. Level density and photon strength functions 1.1 Level density At low excitation energies a large spacing between individual nuclear levels occurs and it is possible to experimentally distinguish these levels with a high accuracy. For rare-earth deformed nuclei, the nuclear level properties (spin, parity, decay properties...) are assumed to be experimentally known up to approximately MeV. With higher excitation energies the level density (LD) increases and the situation becomes more complicated, therefore statistical models are used. However, there exists one additional excitation energy region where nuclear levels can be observed an interval above the neutron separation energy. Neutron resonances in this region can be detected thanks to the neutron time-of flight method. Using a method adopted from statistical physics, Bethe in 1936 [3] derived a formula describing nuclear level density. He assumed that to obtain the complete information on the spectrum of a system, only knowledge on the partition function of the system was needed. Considering a gas of non-interacting fermions with equally spaced non-degenerate single particle levels with fixed spin, the excitation energy dependence of the nuclear level density ρ(e) has the form: ρ(e) = exp(2 ae) 4 3E (1.1) where a is the level density parameter and it differs from nucleus to nucleus. Instead of using LD in the form (1.1), in our simulations of electromagnetic decay two more realistic phenomenological models are tested: Back-Shifted Fermi Gas and Constant Temperature Model. Back-Shifted Fermi Gas (BSFG) model [4] is an extension of the Bethe model described above and it takes into account two types of particles involved. In addition, fermions tend to form pairs, which need extra amount of energy to be separated. We denote this extra energy as an energy shift and the derived formula for the nuclear level density is: ρ(e, J, π) = f(j)f(π) exp(2 a(e )) 12 2σ 1/4 (E ) 5/4 (1.2) with f(j) being the spin and f(π) the parity distribution factors. If the Gaussian distribution of spin projections M is assumed [5], then f (J ) can be expressed as: ( f(j) = exp J 2 2σ ) ( exp ) (J + 1)2 2J + 1 ( exp 2σ c 2σ 2 ) (J + 1/2)2 2σ 2 (1.3) where σ c is the spin cut-off parameter and according to [5] it is taken in the form: σ 2 =.888 a(e )A 2/3 (1.4) 5

10 The parameters a and differ for each nucleus and can be adjusted through fitting experimental data on the level density formula at low excitation energies and just above the neutron separation energy. Later, another form of spin cut-off parameter was proposed [6]: σ 2 =.146A 5/ a(E ) (1.5) 2a No dependence on parity is usually assumed in rare-earth nuclei. This corresponds to f(π)=1/2. The second model was introduced by Gilbert and Cameron [5], who describe the level density by an exponential function, the so-called Constant Temperature Formula (CTF): ρ(e, J) = f(j) ( ) E T exp E (1.6) T with E and T - the nuclear temperature - taken as the free parameters. In this case, the spin cut-off parameter is proposed to be independent of excitation energy [7]. σ = (.98 ±.23)A (.29±.6) (1.7) Von Egidy and Bucurescu introduced a new form of the spin cut-off parameter used in both BSFG and CTF formula [8]: σ 2 =.391A.675 (E.5P a ).312 (1.8) with re-evaluated level density parameters (a, and E, T, respectively). P a refers to deuteron pairing energy calculated from the masses M (A,Z ) of nuclei with proton number Z and nucleon number A. P a = 1 [M(A + 2, Z + 1) 2M(A, Z) + M(A 2, Z 1)[ (1.9) 2 Experimental data on level density come from two types of experiments: The first one employs reactions induced by neutrons, protons and alpha particles to analyse the shapes of neutron evaporation spectra. Within the Hauser-Feshbach statistical approach this method yields absolute values of the level density. A reasonable agreement with BSFG model for a large set of nuclei was achieved [9 12], except for several closed-shell A 28 nuclei where CTF model seems more appropriate. Another kind of experiment, the so-called Oslo Method [13 15] provides information on the level density as well as on the photon strength functions by detecting γ-ray spectra resulting from the charged-particles induced reactions. Mainly ( 3 He, γ) and ( 3 He, 3 Heγ) reactions were used for rare-earth nuclei. However, this method does not give absolute value of the level density, it only describes the shape of the level density energy dependence. Experimental results are in good agreement with the BFSG model for nuclei with A 17. Energy dependence of the level density for two models and two different parametrizations used in the analysis is plotted in Fig In the case of the BSFG model, parameters a = MeV 1 and =.26 MeV for the 6

11 first parametrization and a = MeV 1 and =.39 MeV for the second parametrization were used. For the CTF model parameters E =.62 MeV and T =.6 MeV for the first parametrization and E =.38 MeV and T =.58 MeV for the second parametrization were used B S F G 1 B S F G 2 C T F 1 C T F 2 ρ [M e V -1 ] E [M e V ] Figure 1.1: The energy dependence of the level density for spin J = Photon strength functions When a nucleus undergoes an electromagnetic transition from an initial state α characterized by spin J α and parity π α to a final state β with spin J β and parity π β, a photon of type X and multipolarity L is emitted. This process follows the so-called selection rules: 1. The only contributing transitions are given by Wigner-Eckart theorem: J β J α < L < J β + J α (1.1) Transitions for multipolarities L > 2 are strongly suppressed. 2. Parity selection rules hold: electric multipole operators follow identity ( 1) L π α π β = 1 magnetic multipole operators follow identity ( 1) L+1 π α π β = 1 The probability of the transition from the initial state α with spin J α to the final state β with spin J β through emission of a photon of type X, multipolarity 7

12 L and energy E γ = E α E β per unit time is determined by the partial γ-decay width given by the formula: Γ (XL) αγβ ( ) 2L+1 8π(L + 1) Eγ = B(XL) (1.11) L[(2L + 1)!!] 2 c where the reduced transition probability for deexcitation of the nucleus B(XL) is defined as: B(XL) = 2J β + 1 2J α + 1 B(XL) = 1 2J α + 1 αj α H (XL) βj β 2 (1.12) where we have introduced the reduced transition probability for photoexcitation B(XL), H (XL) stands for electromagnetic transition operator of type X and multipolarity L. When the level density of initial states α becomes significantly high, we use the average partial radiation width: Γ (XL) (E α, I α, π α E β ) = f (XL) (E α E β )Eγ 2L+1 ρ(e α, J α, π α ) (1.13) The function f (XL) is usually called the photon strength function (PSF) and in the single particle approach is assumed to be independent of the transition energy E α E β. From (1.11) and (1.13) numerical relations between reduced transition probabilities B(XL) and photon strength functions f (XL) can be simply derived. For the lowest multipolarity transitions to the state with zero spin it holds: 1 B(E1) [e 2 fm 2 ] = f (E1) [MeV 3 ] (1.14) 1 1 B(M1) [µ 2 N] = f (M1) [MeV 3 ] (1.15) B(E2) [e 2 fm 4 ] = f (E2) [MeV 5 ] (1.16) Here refers to an energy interval summed over. There is a relation between partial radiation width Γ (XL) αγβ of the electromagnetic transition and the XL component of the photoabsorption cross section σ (XL) βγα, which is a result of the detailed-balanced principle: Γ (XL) αγβ = E2 γ (π c) 2 2I β + 1 2I α + 1 σ(xl) βγα (1.17) In agreement with the idea that collective vibrations are independent of the intrinsic nuclear motion, Brink [16] has made an assumption that photoabsorption cross section does not depend on the target state properties (excitation energy, spin, parity) and depends only on the γ-ray energy E γ = E α E β. This so-called Brink hypothesis, see Fig. 1.2, can be extended for any type and multipolarity of the radiation. Although we cannot be sure that the hypothesis applies precisely, 8

13 at this time it seems to be a reasonable description of reality. Supposing the validity of the Brink hypothesis, an alternative definition of the photon strength function is obtained: f (XL) = 1 σ (XL) tot (E β E α ) (π c) 2 (2L + 1)E 2L 1 γ (1.18) where σ (XL) tot is an energy smoothed XL component of the total photoabsorption cross section. Figure 1.2: An illustration of the Brink hypothesis for photoexcitation and γ- decay following the neutron capture reaction [2] Fluctuation properties of partial widths Experimental data show that partial radiation widths Γ (XL) αγβ exhibit strong fluctuations, which can be described by the Porter-Thomas distribution [17]. Probability density function for this distribution is: P (x)dx = 1 2πx e x 2 dx (1.19) with x = Γ αγβ Γ αγβ, see Eq. (1.13). In fact, the Porter-Thomas distribution is a special case of χ 2 distribution with the number of degrees of freedom ν = E1 photon strength function In 1947 Baldwin and Klaiber [18] observed a broad resonance in photoabsorption cross section above the neutron separation energy. As a matter of fact, this resonance had already been predicted a few years before [19] and had been explained by means of collective vibrations of proton and neutron fluids with mutually opposite phases induced by the electric field of the incoming radiation. Because of its dipole character, this motion was named Giant Dipole Electric Resonance (GDER). There are two classical models of nuclear motion characterizing GDER: The first model assumes vibration of proton and neutron fluid within a fixed surface, 9

14 see Fig. 1.3 (a) and the restoring force is proportional to the density gradient of those fluids. In the second one, see Fig. 1.3 (b), the GDER represents vibrations of two incompressible proton and neutron spheres against each other, in this case the restoring force is proportional to the nuclear surface area. For the first model, the resonance position follows A 1/3 dependence [19], [2], while the center of second resonance shifts as A 1/6 [21]. Since the position is well characterized by the dependence E G = 31.2A 1/ A 1/6 MeV, the GDER can probably be described as a mixture of these two models. Figure 1.3: Two models illustrating GDER motion [2]. The most often used phenomenological model of E1 PSF is the Standard Lorentzian model (SLO) (also known as Brink-Axel model). If the Lorentz form of σ (XL) tot in Eq. (1.18) is assumed, then the formula for E1 PSF has the form: f (E1) SLO (E γ) = 1 3(π c) C E γ Γ G 2 Σ (Eγ 2 EG 2 )2 + EγΓ 2 2 G (1.2) where E G and Γ G represent the position and the full width at half maximum of the resonance. The normalization constant C Σ is usually obtained from fitting the photonuclear data and can be generally expressed in the form: C Σ = σ G Γ G (1.21) where σ G is the cross section in the maximum of the resonance. The Lorentz shape of PSF well describes the photonuclear data in the region around the maximum of GDER, particularly at energies E γ 1 MeV. In the case of nuclei with static quadrupole deformation the GDER splits, which can be illustrated within the hydrodynamical model. The shape of GDER is linked to the properties of surface oscillations of the nucleus at the ground state, the intensity of the oscillations along the symmetry axis to those perpendicular to it are predicted to be in ratio 1:2. The experimental photonuclear data support this concept. As a result, E1 PSF is represented by incoherent superposition of two Lorentzian functions from Eq. (1.2) and consequently two sets of Lorentz parameters E G, Γ G and σ G. Intensities of primary transitions following the reaction (n, γ) show inadequacy of the simple Lorentzian shape at energies close to the neutron separation energy for spherical nuclei. On the contrary, for deformed nuclei a reasonable agreement was achieved. In 1983 S.G. Kadmenskij, V. P. Markushev and V. I. Furman [22] proposed another E1 PSF model within the framework of semi-microscopic shell model approach together with the theory of finite Fermi systems. The photon strength function in this so-called KMF model depends not only on the photon energy 1

15 E γ, but also on the excitation energy E f of the nucleus, which is related to the nuclear temperature T f of the final state. Formula for f (E1) KMF at E γ E G in spherical nuclei has the form: f (E1) KMF (E γ, T f ) = 1 3(π c) 2 F Kσ G Γ G E G Γ G (E γ, T f ) (E 2 γ E 2 G )2 (1.22) where the damping width Γ G (E γ, T f ) is related the to nuclear temperature: Γ G (E γ, T f ) = Γ G (E 2 EG 2 γ + 4π 2 Tf 2 ) (1.23) and the nuclear temperature T f depends on excitation energy E f of the final state: Ef p T f = (1.24) a where p is a pairing correction [23] and a is the level density parameter from Eq The authors of [22] propose the value F K.7. Experimental values of E1 strengths obtained from primary transitions to the low-lying energy levels in the reaction (n, γ) have been compared with the KMF predictions for several spherical nuclei [24] and a reasonable agreement has been achieved. It should be noted that this model is suitable only for describing the low energy-region E γ E G as the expression (1.22) diverges at E γ = E G. Later, Kopecky and Chrien [26] suggested the General Lorentzian Model (GLO) in order to set a simple formula for the whole energy range in spherical nuclei. This model combines both approaches mentioned above, for energies E γ E G it reproduces the shape of the KMF model, while near the maximum of GDER it follows the SLO model: f (E1) GLO (E γ, T f ) = [ 1 3(π c) σ GΓ 2 G E γ Γ G (E γ, T f ) (E 2 γ E 2 G )2 + E 2 γγ 2 G (E γ, T f ) +F K ] Γ G (E γ =, T f ) (1.25) It is worth mentioning that the limit E γ for T f is non-zero. This fact is required in order to reproduce experimental data from analysis of 143 Nd(n, γα) reaction [27]. This reaction has been remeasured later with notably better statistics with the result that for low energy region M1 transitions dominate over E1 [28]. Therefore the finite limit of E1 strength for E γ, T f is still not experimentally confirmed. In order to achieve more accurate description of deformed nuclei, a phenomenological modification of the damping width given by Eq. (1.23) has been proposed [29]: Γ G (E γ, T f ) = E 3 G [ k + E ] γ E γ ΓG (1 k ) (E 2 E G E γ EG 2 γ + 4π 2 Tf 2 ) (1.26) where an empirical enhancement factor k was introduced. k stands for enhancement of Γ G (E γ, T f ) at E γ = E γ, value E γ 4.5 MeV is recommended [3]. Obviously for k = 1 we get the relation (1.23). The size of the enhancement has been obtained by simultaneous adjustment of calculated average total radiation 11

16 widths Γ γ, energy dependences of the cross section for (n, γ) reaction and γ-ray spectra produced in this reaction for medium-heavy and heavy nuclei. In our case k is taken as a free parameter. If the damping width Γ G (E γ, T f ) from equation (1.26) is used in the expression for fglo E1 (1.25), one obtains the so called Modified Generalized Lorentzian Model (MGLO). By measurements of radiative capture of light and heavy ions it was confirmed that GDER is a general feature of nuclei up to very high excitation energies. Its position remains stable even in highly excited nuclei and there is only a slight dependence on the excitation energy. This outcome supports the Brink hypothesis in its most general formulation. On the contrary, in our region of interest E γ < S n the amount of experimental information on E1 PSF is not satisfactory. In addition to data mentioned above, i.e. intensities of primary transitions obtained from (n, γ) reaction, alternative source of information are measurements of (γ, γ ) reactions for excitation energies E γ 2 4 MeV [37] and ( 3 He, αγ) and charged -particle induced reactions such as ( 3 He, 3 Heγ) processed by the so-called Oslo method [13], [14], [25] M1 photon strength function Experimental data on M1 strength originate mainly from inelastic scattering using electrons, protons and photons on nuclei and the capture of slow neutrons followed by emission of γ-rays. The amount of information up to neutron separation energy is almost comparable to that on E1 PSF, while for energy region above the particle emission threshold the lack of relevant data is a consequence of the E1 strength being dominant over other multipolarities. Within the Single-Particle Model (SP), which is sometimes used, f (M1) SP is assumed to be independent of the transition energy, i.e. a constant. Estimated values of f (M1) SP can be obtained especially from measurements of primary γ transitions following the resonance neutron capture. As a result of Porter-Thomas fluctuations, partial radiation widths of γ transitions from neutron capturing state to low-lying energy levels have to be averaged over a high number of neutron resonances to get a reasonable estimate of f (M1) SP. Such averaging is used in the socalled Average Resonance Neutron Capture (ARC). Unfortunately, it is difficult to achieve absolute PSF value using this method and a normalization is required. It can be in principle achieved by averaging the partial radiation widths determined for individual resonances using the neutron time-of-flight method combined with the conventional γ spectroscopy [31]. An important outcome from ARC data is the ratio f (E1) /f (M1) = 7±1 at energy E γ 7MeV, which holds for the majority of nuclei with A 1 [31]. Single-particle model mentioned above appears to be too simple, it seems that there exist resonance structures in M1 PSF at energies below or close to S n. One resonance structure seems to be observed throughout the periodic table, it is known as the Spin-Flip resonance (SF). For deformed nuclei a double-humped resonance SF structure was found in inelastic scattering of protons having kinetic energy T p = 2 MeV, the positions of its centers seem to be given by 34A 1 3 and 44A 1 3 [32]. A different parametrization of the SF can be found in [33], [34]: the center center shifts as E SF 41A 1 3 and the width is Γ SF = 4 MeV. In 197 s a collective M1 mode in the framework of a simple geometrical 12

17 Two-Rotor-Model was proposed for deformed nuclei [36]. This M1 mode can be visualized as a scissors-like counterrotation of proton and neutron fluid, for this reason the mode is usually referred to as the Scissors Mode (SC). The presence of strong 1 + excitations was observed a few years later thanks to high-resolution inelastic electron scattering from even-even rare-earth nuclei [35]. In 8 s and 9 s the Nuclear Resonance Fluorescence (NRF), i.e. reaction (γ, γ ) became a successful method for studying the SC mode because of its high sensitivity and selectiveness to the low-lying dipole excitations. A remarkable result of these experimental data was the dependence of the total strength of the SC mode on the square of deformation parameter δ 2 at least for even-even nuclei. In order to reproduce the NRF data the formula for the energy center E SC has been proposed in the form [38]: E SC = δ 2 A 1/3 (1.27) In the rare-earth region, the energy is close to 3 MeV. The total SC strength deduced from (γ, γ ) reactions on well-deformed even-even rare-earth nuclei is approximately 3 µ 2 N. In these nuclei the strength is mostly distributed over a small number of states corresponding to the width Γ SC 2 3 kev. As a consequence of higher level density in odd and odd-odd nuclei and therefore more complicated detection of all the transitions in NRF, the observed strength reported in these nuclei is usually smaller. Successful experimental method for investigating SC mode is the coincidence measurement of photons from neutron capture reactions. This can be measured in two types of experiments: two-step γ-cascades (TSC), where semiconductor detectors with high resolution are used and multi-step γ-cascades (MSC) using scintillator detectors, which is our case. Spectra of TSC cascades following neutron capture in 155,157 Gd were analysed and resonance structures at energy E SC = MeV and E SC = MeV for 156 Gd and 158 Gd, respectively, were found. For damping widths almost any value from interval Γ SC =.5 1. MeV [48] is reasonable. In reality, a resonance structure at E γ MeV was seen earlier in the γ-ray spectra obtained from neutron capture at neutron energies of several tens kev to MeV [39 41] in deformed nuclei. This result demonstrates that the position of the resonance is independent of the incident neutron energy, which indicates that the resonance is present not only in transitions to the ground state, but also in transitions between excited states. It should thus correspond to the photon strength function. The SC mode was observed also in reactions ( 3 He, αγ) and ( 3 He, 3 Heγ). The full width at half maximum obtained here is higher than in (γ, γ ) but comparable to the TSC and the MSC experiments, Γ SC 1. MeV. From experiments with helium probes [42] the value of the cross section in the maximum of the resonance was estimated σ SC =.5 mb, while from the fast neutron experiments [39] σ SC.2.4 mb. The note should be made that Brink hypothesis was assumed in all analyses discussed above. The energy dependence of the E1 and M1 models used in the thesis is shown in Fig Notation (E G ; Γ G ; σ G ) for parameters of the resonances will be used. In the case of the SLO model, sum of two resonance structures in the form (1.2) was used. GDER parameters were (12.9 MeV; 3.66 MeV; mb) and 13

18 (15.54 MeV; 3.99 MeV; mb). KMF and MGLO models depend also on the energy of the final state E f, the dependence is plotted for E f = 2 MeV. Parameters for the KMF model are p = MeV and a = MeV 1. In the case of MGLO model, parameter values are k = 3 and E γ = 4.5 MeV. For M1 PSF model combination SP+SF+SC is plotted. f SP = MeV 3, parameters of the SF mode are (6.2MeV;.8MeV;.8mb) and (7.7MeV; 1.8MeV; 1.2mb) and parameters of the SC mode (3.2 MeV;.6 MeV;.5 mb). Unless stated otherwise, parameters listed above were used in the simulations. 2. x x 1-7 S L O K M F M G L O S P + S F + S C f [M e V -3 ] 1. x x E γ [M e V ] Figure 1.4: Energy dependence of the E1 and M1 PSF models E2 photon strength function In addition to dipole transitions, also the electric quadrupole ones presumably contribute to statistical part of the γ-ray spectra accompanying the compound nucleus reaction. Concerning the decay of an excited nucleus with a low spin, this type of transitions is believed to play only a minor role. The main source of experimental information on E2 strength is inelastic scattering of charged particles [45]. E2 strengths obtained from the inelastic scattering of nuclei with electrons, protons and α-particles [34] indicated resonance behaviour, which was interpreted as the isoscalar Giant Quadrupole Electric Resonance (GQER). This behaviour corresponds to a concept of a surface oscillation of neutrons moving together with protons. The energy center of the GQER resonance E R = 63A 1/3 MeV [46] almost coincides with the centroid of the GDER. Authors of [47] recommend formulas for the damping width Γ R and the cross section at the maximum of the GQER σ R : Γ R [MeV] = A, σ R [mb] = Z 2 ER 2 A 1/3 /Γ R, where Γ R and E R are supposed to be expressed in MeV. 14

19 Up to now only a small number of E2 primary transitions has been observed in (n, γ) reactions. Furthermore, their intensities are biased towards large values due to the Porter-Thomas fluctuations, as only the strongest ones can be observed. Alternatively simple single-particle model is used. E2 strength plays a role in the transitions within the rotational bands in the region of low excitation energies. However, description with photon strength function is not valid at low excitation energies. For higher multipolarities (M2, E3,...) the contribution to γ-spectra is strongly suppressed, hence they are not included in our simulations. Only a several weak transitions of this type have been detected yet, mainly in low excitations region. 15

20 2. Experiment and data processing 2.1 DANCE detector array Figure 2.1: A cutaway view of the DANCE detector [49] The measurement was carried out at the neutron source at LANSCE. By picking up an extra electron from a Cs layer, a current of negative hydrogen ions is produced and accelerated with electrostatic lens to energy 75 kev. The beam further obtains energy of 8 MeV in a linear accelerator, is stripped by a thin foil to H + and then injected into the Proton Storage Ring (PSR). The average proton current is 1 µa and the beam is transformed into 125 ns wide pulses. Proton beam is extracted from the PSR with a repetition rate of 2 Hz and strikes a tungsten spallation target placed in the Manuel Lujan Jr. Neutron Scattering Center. The resulting neutrons have energies from thermal up to several MeVs. In order to enhance the low-energy part of the neutron flux, the target is surrounded by a water-moderator block and by Be and Pb reflector. The neutrons are sent to a 2.3 m flightpath to the DANCE detector, see Fig The neutron energy corresponds to the time of flight, which differs from hundreds of nanoseconds for neutrons with energies 2-2 MeV to 14 milliseconds for 1 mev neutrons. There are three detectors monitoring the neutron beam: A 6 LiF target is positioned on a kapton foil in the center of the beam pipe at the angle of 45, about m from the neutron moderator. The neutron flux is measured via reaction 6 Li(n, α) 3 H, where α-particles and triton are detected with two n-type surface barrier Si detector placed perpendicular to the beam, 3 cm far from the 6 Li foil. A proportional chamber filled with BF 3 + Ar gas, located m from the neutron moderator. The corresponding reaction is 1 B(n, α) 7 Li. 16

21 Figure 2.2: Flight path FP14 in ER-2 experimental area at Manuel Lujan Jr. Neutron Scattering Center LANSCE [51]. A fission chamber detecting fragments from the 235 U(n, f) reaction. There are several collimators arranged in the flightpath. As the neutron beam diverges with an increasing distance from the last collimator, the neutron flux measured by above mentioned detectors is thus not the same as the flux in the center of the DANCE where the target is located. For this reason, to obtain information on the neutron beam profile and normalization, gold activation has been used. The diameter of the neutron beam at the entry to the DANCE detector is 1 cm, while at the BF 3 monitor it is approximately 2.5 cm. The DANCE detector is a highly efficient, almost 4π γ-ray calorimeter consisting of 16 BaF 2 crystals that is designed for studying neutron capture cross sections on small samples. A full 4π array is composed of 162 crystals, two are left out to create space for the neutron beam tube. The crystals are constructed in four different shapes, see Fig. 2.3, and are arranged to compose a BaF 2 sphere of an inner radius 17 cm and a thickness of 15 cm. In order to absorb neutrons that scatter from the sample and can possibly strike the crystals, a 6 cm thick 6 LiH shell is placed between the sample and the crystals, where the neutrons are attenuated due to the reaction 6 Li(n, α). The crystals are surrounded by an aluminium spherical structure with an inner radius of 49.7 cm and an outer radius of 53.5 cm. In addition, each crystal is covered in a.7 mm thick PVC foil and glued to a photomultiplier tube. Figure 2.3: The DANCE detector consists of BaF 2 scintillation crystals with 12 of them being regular pentagons of type A and three different shapes of hexagons, namely 6 regular crystals of type B, 6 of type C and 3 of type D [51]. 17

22 Isotope Abundance [%] 162 Er Er Er Er Er Er.33 Table 2.1: Isotope composition of the Er target. The efficiency for detecting a single 1 MeV photon is approximately 86 %. In a typical γ-ray cascade accompanying the radiative neutron capture 3 or 4 γ-rays are emitted, then the total efficiency exceeds 95 %. The 2.1 mg Er target with size.25 x.25 inches was positioned between two layers of polyester, each 1.5 µm thick. The isotopic composition is listed in Tab Data acquisition The scintillation signal from the BaF 2 crystals has two components: the fast one with decay time τ 6 ps, wavelength 22 nm and the slow component τ 6 ns, wavelength 31 nm - see Fig and their intensities are collected independently. The photomultiplier output pulses for each crystal are digitized in Acqiris 4- channel DC256 digitizers with 8-bit resolution and sampling rate of 5 MHz (i.e. 2 ns sampling interval). The electronics allowed to reload the crystal waveforms in 5 µs long window after each neutron pulse. This acquisition time window can be arbitrarily delayed with respect to the proton beam pulse. In our case the 5 µs window covered all neutron resonances with neutron energies E n 9 ev. In order to reduce the amount of data, the waveforms are processed online before storing. Each waveform is divided into several sections, see Fig. 2.2, and the following information is obtained: (1) the pre-peak integral of a background baseline reduced to a 1 ns wide integral, (2) the fast component consisting of 32 data points (i.e. 64 ns because of 2 ns sampling rate), (3) the slow component composed of five 2 ns wide sequential integrals and (4) two time stamps (relative to the beam pulse and master clock). In this way the amount of data sent to the central server is less than 1 MByte/s. Before the arrival of the next proton beam pulse, the data acquisition system has approximately 4 ms to read out the digitizers, get the necessary waveform information and send the data to a MIDAS Server (Maximum Integrated Data Acquisition System). To achieve this, the digitizers are placed into 14 compact PCI crates, each crate containing 6 4-channel digitizers and handling 12 BaF 2 crystals. 18

23 Figure 2.4: Gamma-ray and alpha-particle signal from a BaF 2 crystal [49]. Figure 2.5: The signals are characterized by storing the fast component and the time of the first data point. The slow component consists of five equal width integrals [49]. 19

24 2.3 Detector granularity The DANCE detector contains 16 BaF 2 crystals and its high granularity can be used to analyse the number of crystals that fire. Interaction of γ-rays with energies exceeding several hundreds of kev goes primarily via Compton scattering and for higher energies by production of electron-positron pairs. Hence often a single γ-ray does not deposit its full energy in a single crystal and the number of crystals that fire is higher than the true multiplicity of the event. In a cascade following the neutron capture event several γ-rays are usually emitted, whether it is caused by the neutrons captured in the target or induced by the scattered neutrons in the crystal. However, there is a certain pattern in the number of crystals that fire. Scattered neutrons captured in a specific crystal create several γ-rays that probably hit adjacent crystals. On the contrary, numerous gammas following the neutron capture in the target most likely fire crystals located in different regions of the detector. Nonetheless, this method is not used in our further analysis. A cluster was defined as a set of neighbouring crystals that are hit and it was shown [5 53] that cluster multiplicity - using the definition of cluster mentioned above - copies the real multiplicity of gamma cascade better than the crystal multiplicity. 2.4 Time alignment and energy calibration Time alignment of individual crystals was needed, since time delays of the signals from different crystals can deviate. By using the average time difference between the events in the crystal and the reference crystal, these deviations were corrected run-by-run. In Fig. 2.6 time difference of the signals for one crystal with respect to the reference crystal is shown. As illustrated, the alignment within a few nanoseconds can be achieved - therefore optimal coincidence window seems to be about 1-2 ns. Within this interval, all coincidences are taken into account and there are only a few random coincidences. Widening the interval would lead to more random coincidences and increase of the background. Unless stated otherwise, coincidence window 2 ns was used in the analysis. The energy calibration of the BaF 2 crystals was done by measuring γ spectra from the natural radioactive sources 22 Na (.511 and MeV), 6 Co (1.173 and MeV) and 88 Y (.898 and MeV), arranged at the target position. Since Ba and Ra are chemical homologues, BaF 2 crystals always contain radioactive isotopes from the radium decay chain, see Sec α-particles which are emitted can be easily distinguished, because the intensity ratio between the slow and the fast components of the signal is significantly different for γ-rays and α- particles, see Figure 2.4. Due to temperature changes in the scintillators, there were small gain shifts detected in each crystal during the running of experiment, thus run-by-run energy calibration was carried out. Exact gains and shifts to individual runs were applied using α-decay energy measured in each crystal during the experiment. The α-particle peak positions, see Fig. 2.7, were fitted for each run and each crystal. 2

25 Time calibration Counts t [ns] Figure 2.6: Time difference t of the signals from the first crystal with respect to the reference crystal. Figure 2.7: Left: Decay chain of 226 Ra being homologue to Ba in BaF 2 crystals (energies of α-particles in MeV). Right: α-particle spectrum from a single crystal used for energy calibration. Peaks correspond to α-particle energies from the decay chain on the left (4.78 MeV, 5.49 MeV, 6. MeV and 7.69 MeV). 2.5 Experimental sum-energy and MSC spectra The stored information on photon energy was sorted using the coincidence window and sequences {i, M, E γ (1), E γ (2),..., E γ (M) } were prepared. Here i refers to neutron energy region, M to cluster multiplicity and E γ (k) to energy of the k th cluster. In Fig. 2.8 the neutron time-of-flight spectrum for cluster multiplicity M =3 is shown. With the help of above mentioned information, events corresponding to individual neutron resonances are chosen. They are used to construct the spectra of γ-ray energy sums E Σ for individual multiplicities, i.e. spectra of total deposited energy. These sum-energy spectra for three different neutron resonances are shown in Fig Each sum-energy spectrum consists of: 21

26 Counts Neutron time-of-flight spectrum Neutron energy [ev] Figure 2.8: Neutron time-of-flight spectrum for cluster multiplicity M =3. A full-energy peak corresponding to the detection of full energy available from the neutron capture reaction, it is located close to the neutron separation energy S n. A low-energy tail related to the γ-cascades for which a part of the emitted energy was not detected in the array. In our case also peaks at energies approximately 1.1 and 6.5 MeV are visible, especially for multiplicities M =1-4, they corresponds to γ-cascades going via an isomeric state at excitation energy E = 1.94 MeV. This isomeric state has its lifetime τ = 19 ns [54], while lifetimes of other states in this nucleus is in orders of ps. When the nucleus decays via this level, because of its lifetime being much longer than the coincidence window 2 ns, in some cases one γ-cascade is detected as two separate events. The first one corresponds to the part of the cascade feeding the isomer - it has E Σ 6.5MeV, while the second part of the cascade comes from decay of the E Σ = 1.94 MeV level. The peak near 6.5 MeV is clearly visible for M =2-4, the peak near 1.1 MeV is visible mainly in M =1-3 - it is a consequence of the isomeric decay of the state which goes via one or two γ. In fact, the position of the full-energy peak is slightly shifted down with respect to S n - the difference is approximately 3 kev - as a result of internal electron conversion for low-energy transitions in the product nucleus. The shape of the sum-energy spectra can be influenced by the experimental background, for detailed description see Sec For low neutron energies only s-wave neutron capture plays a major role, therefore only J π = 3 + and J π = 4 + neutron resonances are observed. In Fig. 2.1 the sum-energy spectra for both spins of neutron resonances are shown. It is clear that their shapes are different, there is especially a shift in multiplicity distribution. This difference can be used for determination of neutron resonance spin, for more details see e.g. [55]. We have found that spins of all strong resonances are consistent with available information in literature [56]. 22

27 M = M = 3 4 E Σ [M e V ] 1 5 E Σ [M e V ] M = e V e V e V M = 5 E Σ [M e V ] E Σ [M e V ] Figure 2.9: Experimental sum-energy spectra from three J π = 3 + neutron resonances for cluster multiplicities M =2-5. From the sum-energy spectra the so-called multi-step cascade spectra are constructed. For their construction only events contributing to the full-energy peak in the sum-energy spectra are chosen. The considered energy range in the sumenergy spectra used for construction of the MSC spectra - Q-value range - is denoted as Q, in our case Q= MeV. Widening of this range would lead to better statistics, but also to suppression of the structures observed in the spectra. Narrowing has practically no influence on the shape of the spectra and only causes reduction of the statistical precision. The MSC spectrum for multiplicity M refers to a spectrum of energies associated with a single cascade that are deposited in M detector clusters. In Fig MSC spectra for three different neutron resonances with spin J π = 3 + are shown. Spectra for resonances with two different spins are shown in Fig The shape of the spectra with the same resonance spin is nearly identical, for different spins again a shift in the multiplicity distribution is observed. As mentioned in subsection 2.3, cluster multiplicity is closer to the true multiplicity of the γ-cascade than the crystal multiplicity. In addition, the majority of the events in the spectra are for cluster multiplicities M =2-5, therefore the spectra are drawn for these multiplicities. M =1 sum-energy spectrum is strongly influenced by the decay of the isomeric state, see Fig. 2.13, and for M > 5 the statistics is small. All spectra are normalized to the total number of events for multiplicity M = 3 in the MSC spectra. 23

28 e V, J π = e V, J π = 3 + M = M = E Σ [M e V ] 2 5 E Σ [M e V ] M = 4 M = 5 E Σ [M e V ] E Σ [M e V ] Figure 2.1: Experimental sum-energy spectra from J π = 4 + and J π = 3 + neutron resonances for cluster multiplicities M = M = M = E γ [M e V ] M = E γ [M e V ] e V e V e V M = 5 E γ [M e V ] E γ [M e V ] Figure 2.11: Experimental MSC spectra from three J π = 3 + neutron resonances for cluster multiplicities M =

29 M = M = E γ [M e V ] E γ [M e V ] e V, J π = e V, J π = M = 4 M = 5 E γ [M e V ] E γ [M e V ] Figure 2.12: Experimental MSC spectra from J π = 4 + and J π = 3 + neutron resonances for cluster multiplicities M = M = e V e V e V E Σ [M e V ] Figure 2.13: Experimental MSC spectra from three J π = 3 + neutron resonances for cluster multiplicity M =1. 25

30 2.6 Influence of parameters on MSC spectra Coincidence window We can test influence of various parameters on the resulting experimental spectra, one of them is the width of the coincidence window τ. In Fig sum-energy spectra for τ=8, 2, 5, 1 and 2 ns are shown. Spectra for τ=8 and 2 ns are nearly identical. The difference between the spectra for τ 5 could be expected as these coincidence windows are comparable to the mean lifetime of the 1.94 MeV isomeric state. The decrease at 1.1 and 6.5 MeV peaks nicely corresponds to cascades via this isomeric state. In Fig MSC spectra for coincidence windows 8, 2, 5, 1 and 2 ns are shown, for τ 5 ns the difference between the shapes of the spectra is very small. This means that the shape of MSC spectra is not affected by our choice of the coincidence window for τ 5 ns, for higher value of τ the shape is influenced by random coincidences. Such a stability is perfect, as our results on MSC spectra do not depend on the exact choice of τ M = 2 M = 3 E Σ [M e V ] E Σ [M e V ] 2 8 n s n s 5 n s 1 1 n s 5 2 n s M = 4 M = 5 E Σ [M e V ] E Σ [M e V ] Figure 2.14: Experimental sum-energy spectra from J π = 3 +, E n = 39.4 ev neutron resonance for different values of coincidence window Detection threshold Another parameter that can be checked is minimum detectable photon energy, i.e. the detection threshold E thr, which is slightly different for each detector due to imperfections in energy calibration. Experimental spectra for values of E thr =1, 13, 16 and 2 kev are visualized in Fig The similarity of the 26