Feb 3, 2018 - Neutron and proton reactions at 20 and 80 MeV have been considered. .... 3.3.3. Comparison with Modified Williams' Formula. 103. 3.3.4. ... The initial condition for solving the master equation. DWBA .... Velocity of the emitted particle. ... The nuclear reaction terminates when the compound nucleus rests by.
Ministry of Higher Education and Scientific Research
وزارة ا ا وا ا
University of Baghdad
آـ اـــ م
ـ ﻡـ ﺏــاد
College of Science
Neutron Induced Preequilibrium Nuclear Reactions Using the Exciton Model A Thesis Submitted to the Department of Physics College of Science, University of Baghdad In Partial Fulfillment for the Requirements of the Degree of
Doctor of Philosophy in Physics
by
Ahmed Abdul-Razzaq Selman B.Sc. in Physics 1997, M.Sc. in Physics 2000
Prepared Under the Supervision of
Dr. Mahdi Hadi Jasim
March 2009
Dr. Shafik Shaker Shafik
Rabee'a Alawal 1430 Hejri
To My Parents…
Ahmed
Supervisors Certification We certify that the thesis titled “Neutron Induced Preequilibrium Nuclear Reactions Using the Exciton Model” was prepared by the Ph.D. student
Ahmed Abdul-Razzaq Selman under our supervision at the Department of Physics, College of Science, University of Baghdad, as a partial requirement for the Degree of Doctor of Philosophy in Physics.
Dr. Mahdi Hadi Jasim
Dr. Shafik Shaker Shafik
Department of Physics, College of Science, University of Baghdad
Department of Physics, College of Science, University of Baghdad
/ / 2009
/ / 2009
In view of the available recommendation, I forward this thesis for debate by the examination committee.
Prof. Dr. Baha T. Chiad Chairman of the Department of Physics, College of Science, University of Baghdad
/ /2009
ii
Examination Committee Certification We have examined the Ph.D. student, Ahmed Abdul-Razzaq Selman, in the material presented in his thesis titled “Neutron Induced Preequilibrium Nuclear Reactions Using the Exciton Model” and found it adequate for Ph. D. Thesis with (very good) grade.
Signature: ............................................... (Chairman) Name: Prof. Dr. Ra'ad A. K. Radi Title: Professor in Theoretical Nuclear Physics Date: / / 2009
Signature: ............................................... (Member) Name: Prof. Dr. Hazim L. Mansour Title: Professor in Applied Nuclear Physics Date: / / 2009
Signature: ............................................... (Member) Name: Dr. Adel K. Hamoodi Title: Assistant Professor in Theoretical Nuclear Physics Date: / / 2009
Signature: ............................................... (Member) Name: Nada F. Tawfiq Title: Assistant Professor in Applied Nuclear Physics Date: / / 2009
Signature: ............................................... (Member) Name: Dr. Nadia M. Adeeb Title: Assistant Professor in Theoretical Nuclear Physics Date: / / 2009
Signature: ............................................... (Supervisor and Member) Name: Dr. Mahdi H. Jasim Date:
/ / 2009
Signature: ............................................... (Supervisor and Member) Name: Dr. Shafik S. Shafik Date: / / 2009
Approved by the Dean of the College of Science, University of Baghdad. Signature: ............................................... The Dean Prof. Dr. Khalid S. A. Al-Mukhtar Date: / / 2009 iii
Acknowledgment I
would
like
first
to
show
my
gratitude
to
my
supervisors,
Dr. Mahdi Hadi Jasim and Dr. Shafik Shaker Shafik for their fruitful and continuous discussions during this work. Their encouragement and trust were one of the guides without which this work could not reach this level. The efforts of the Chairman and staff of the Department of Physics are kindly appreciated. My special thanks are for Dr. Khalid S. A. Al-Mukhtar, the Dean of the College of Science, University of Baghdad, for his great support and encouragement. My most appreciation and gratitude goes to my family for their endless and great help during my study.
iv
Abstract The statistical exciton model of nuclear reactions has been used to study and calculate Preequilibrium Emission (PE) spectra and reaction cross-sections at intermediate energies. Numerical calculations have been made for a twocomponent system, for (N,N) reactions with
54,56
Fe,
103
Rh, and
209
Bi nuclei.
Emission spectra and reaction cross-sections have been numerically calculated, and compared with experimental results. Two important contributions to the existing model are made in the present work, namely: 1- The development of a complete and analytical solution to calculate the state density of excited nuclear states based on the one-component non-Equidistant Spacing Model (non-ESM). Beside the uncorrected case, two main corrections are considered in this formula, namely, Pauli blocking energy and pairing effect. Analytical and numerical comparisons have been made with three of the earlier formulae showed that the present formula gives a general solution. Numerical comparison with exact calculations showed that the current results are still higher than exact calculations, which reflects the fact that more corrections are required to be included in this treatment. 2- The development of a new and simple numerical approach to solve the master equation. Comparisons with earlier methods showed that the current development can be implemented during practical calculation at energies less than~80 MeV. Neutron and proton reactions at 20 and 80 MeV have been considered. The results showed that the accuracy of this treatment is within ~5% of error compared with other methods. A variety of quantities required during PE calculation, such as the state density of the excited states, the effective matrix element of the specified reaction, the occupation probability of exciton states, and others have been studied and thoroughly discussed, and numerically calculated using a library of MATLAB codes written for this purpose. Beside the above, various correction added to the state density calculations have been studied. These corrections include: Pauli correction term, pairing effect, v
Finite Well Depth (FWD) and Back-Shifted Fermi Gas (BSFG) correction, surface effect, spin, isospin, angular and linear momentum distributions, active and passive hole correction and charge effect. It is found from these results that, as the excitation energy increases the state density values increase, which is attributed to the availability of more degree of degeneracy for each state. Furthermore, as more corrections are added, the calculated value decreased, and this is explained due to the decrement in the effective excitation energy. Other model parameters such as the reaction strength and type of internal scattering reaction involved during exciton development are also discussed and calculated. PE spectra for selected reactions were numerically calculated and compared with earlier calculations. Specially for
54
Fe(p,p), and
103
Rh(p,n) reactions, earlier
calculations could be exactly regenerated. Similar remarks were obtained for 56
Fe(n,n/) reaction at incident neutron energy 20 MeV, and for 103Rh(n,n/) reaction
at 18.8 MeV. 103Rh(p,p) reaction cross-sections has also been calculated. In order to compare with experimentally measured emission spectra and cross-sections, other emission components must be included such as evaporation and nucleon transfer components. These components are calculated and added to the PE component, then the resultant spectra were compared with experimental data resulting a better agreement with experiment. State density effects are discussed during these comparisons, and it is seen that the most important effect is due to the dependence of the single-particle level density, g, on energy. The calculated emission spectra and reaction cross-sections approved good match with experiment at intermediate energies.
vi
Contents Dedication Supervisors Certification Examination Committee Certification Acknowledgments Abstract Contents List of Alphabetic Symbols List of Greek Symbols
i ii
Chapter One: General Introduction and Review
1
1.1 . Introduction 1.2 . Experimental Evidences 1.3 . Literature Review 1.3.1. The Preequilibrium Models Development and Applications 1.3.2. Reactions Involving γ-rays 1.3.3. State and Level Density Calculations 1.3.4. Angular Distribution of PE and Kalbach Systematics 1.3.5. The Master Equation 1.4 . Aim of the Present Work
1 5 8 9 14 15 17 18 20
Chapter Two: Theory of the Exciton Model
21
2.1. Introduction 2.2. The Exciton Model: 2.2.1. A general Description 2.2.2. Basic Assumptions 2.2.3. Emission Rates and the Principle of Detailed Balance 2.3. The State Density 2.3.1. Ericson's Formula 2.3.2. Williams' Formula 2.3.3. Pairing, and Modified Williams' Formula 2.3.4. Finite Well Depth Correction and Back-Shifted Levels 2.3.5. Surface Effect and Passive and Active Holes 2.3.6. Charge Effect 2.3.7. Spin and Angular Momentum Distribution 2.3.8. Isospin Dependence 2.3.9. non-ESM Formula 2.4. The Occupation Probability 2.5. The Master Equation 2.5.1. Solution Methods 2.5.2. The Present Method 2.6. Transition Rates
21 21 21 23 26 28 31 32 33 35 39 42 43 45 46 49 51 52 55 57
iii iv v vii ix xii
vii
2.7. The Emission Spectrum 2.8. Kalbach Systematics
59 60
Chapter Three : Results and Discussions
64
3.1. Introduction 3.2. State Density Results 3.2.1. Ericson’s Formula 3.2.2. Williams’ Formula 3.2.3. FWD Correction 3.2.4. Pairing Correction With and Without BSFG 3.2.5. Surface Effect 3.2.6. Angular and Linear Momentum Distribution 3.2.7. Comparison with PLD Code 3.3. The State Density Results for non-ESM Formula 3.3.1. Comparison with Ericson’s Formula 3.3.2. Comparison with Williams’ Formula 3.3.3. Comparison with Modified Williams’ Formula 3.3.4. Comparison with Exact Calculations 3.4. The Master Equation Results 3.5. The Transition Rates 3.6. The Decay Rates 3.7. The Mean Lifetime of the Exciton States 3.8. The Emission Spectrum 3.8.1. Pure PE Spectra 3.8.2. PE Spectra with Evaporation and NT Components 3.9. Reaction Cross-Section Calculations
64 64 65 68 71 74 82 85 90 94 95 99 103 107 107 114 124 129 132 132 135 146
Chapter Four: Conclusions and Future Work
149
4.1. Conclusions 4.2. Future work
149 153
References
154
viii
List of Symbols List of Alphabetic Symbols Symbol a A (or AA) AB Ap,h (or Apπ,hπ,pν,hν) Aa Ab
Description The level density constant. The mass number of the target nucleus. The mass number of the residual nucleus. Pauli blocking energy for one-component (or two-component) system.
AMD bq B Bp,h (or Bpπ,hπ,pv,hv) B(p,h,λ)
The mass number of the incident particle = Za + Na. The mass number of the emitted particle = Zb + Nb. The advanced Pauli correction factor that includes the improved pairing correction (the subscript K stands for Kalbach who suggested this term). Angular Momentum Distribution. Bernoulli numbers of the order q. Binding energy of the nucleon. Modified Pauli blocking energy for one-component (or two-component) system. Numerical coefficients for the state density with BSFG correction.
BROND
The Russian Evaluated Neutron Data Library of general purpose.
BSFG c cn Chj and Cpi Ce CN d dσ/dε d2σ/dε dΩ D(nπ,nν) DWBA E Eeff Ephase Eth Esym ESM EXFOR Evap. f1(n,E,V) fT(p,h,T), f(∆) F FGM FKK
Back-Shifted Fermi Gas model. Speed of light in vacuum. A constant used in the angular momentum distribution function. Numerical coefficients, Cpi =p!/i!(p-i)! and similar equation for Chj. Condensation energy. Compound Nucleus. Energy spacing in the ESM approximation, g=A/d. The differential cross-section (emission spectrum). The double-differential cross-section. The initial condition for solving the master equation. Distorted-Wave Born Approximation. Excitation energy. Effective excitation energy. Pairing energy for phase transition. Threshold excitation energy. Symmetry energy. Equi-distant Spacing Model. EXchange FORmat library, the international experimental data library. Evaporating component. The correction function due to surface effect in the ESM approximation. Correction factor of the isospin quantum number. Correction function of energy that depends on pairing. Fermi energy. Fermi Gas Model. Feshbach, Kerman and Koonin model for PE reactions.
AK (p,h)
ix
FWHM FWD g gπ (or gν) go G(p,h) GDH GDR h hπ (or hν) HM HMB
IAEA J K
Full-Width at Half-Maximum. Finite Well Depth. Average single-particle level density. Single-particle level density for proton (or neutron). Single-particle level at Fermi energy. A correction term for Ap,h. Geometry-Dependent Hybrid model. Giant Dipole Resonance. Hole exciton number for one-component. Hole exciton number for proton (or neutron) particle for two-component. Hybrid Model. Harp, Miller and Berne model. The energy spectrum for a particle of type β, emitted with energy ε after time t from reaction start point. International Atomic Energy Agency. The (total) angular momentum of the target nucleus. The linear momentum.
JENDL
The Japanese Evaluated Nuclear Data Library.
Kproj KF Kπ(n) LMD m M M(p,h,E,K) MCH MSC MSD n nc nmax nπ (or nν) N Na NR, NC NT p pπ (or pν) po , ho , no p, h , n pm pβ
The average square value of the momentum projection on the direction of the linear momentum K. The linear momentum at Fermi surface. The charge factor for proton emission. Linear Momentum Distribution. The nucleon effective mass. The matrix element of the residual interaction. The linear momentum distribution function. Monte Carlo Hybrid model. Multi-Step Compound component. Multi-Step Direct component. Exciton number. Critical exciton number Maximum exciton number. Exciton number for protons (or neutrons) only. n=nπ+nν. neutron number of the target nucleus. Neutron number of the incident projectile. Number of states for residual and compound nucleus respectively. Nucleon Transfer component. Particle exciton number for one-component. Particle exciton number for protons (or neutrons) for two-component. Initial number of p, h, and n, respectively. Most probable numbers of p, h, and n, respectively. Maximum(p,h). Momentum of the emitted particle.
P
Occupation probability.
Iβ(ε,t)
x
The occupation probability of finding a specific state density due to pair creation (or exciton creation) from other states with (p-1) particles. The total occupation probability of finding a specific state density passing P2(p, pπ) through a certain configuration. Pairing correction as a function of pairing energy. P(∆) Legendre polynomial of the ℓth degree, where ℓ is the orbital angular Pℓ(cos θ) momentum q.n. PE Preequilibrium Emission PLD Partial level density. Q(n), Charge factor. ro The classical nucleon radius. R Angular momentum distribution function. R(J) The shape distribution of the total angular momentum, J. Ro The nuclear radius. R(n) A general charge factor. Spin of the emitted (and incident) particles. sβ ( sα) SMMC Shell Model Monte Carlo model. t Nuclear temperature. T Isospin of the incident particle. Tz The z-component of the isospin. The time required for exciton state equilibration. T(nπ,nν) up (or uh), u The single-particle excitation energy for particles (or holes). TUL Tamura, Udagawa, and Lenske model for PE reactions. U Energy of the residual nucleus, U = E – ε – B . Up Pairing correction for the total state densities. Uth Effective Threshold Energy. vo Velocity of the emitted particle. Vnucl. Confinement volume of the nucleus (the nuclear volume). V The nuclear potential depth. V1 The average effective well depth. WRC, WCR Particle emission rates from residual to compound nucleus and vise versa. Particle emission rate, for emitted particle of type β with energy ε from Wβ(p,h,E) exciton state defined by p, h and E. Z Proton (Atomic) number of the target nucleus. Za Proton number of the incident projectile. P1(p, pπ)
xi
List of Greek Symbols Symbol
αp,h β ) Ξ
Description Correction term for state density formula. A subscript denoting the type of the emitted particle.
A special mathematical operator used to replace multiple sums.
δ
∆ ∆n ∆oπ (or ∆ον)
ε εp (or εh)
Γvπ ( Aa + 1, Z a )
η ћ
µβ ν
λx y
D
π
Θ(x)
ρ σ σg σn σρ
σβ(ε) σβ, eq.(ε)
τn ω1,2(p,h,E) ωf(p,h,E)
Dirac delta function. Pairing gap energy. The change in the exciton number. The gap energy for protons (or neurons). Energy of the emitted particle. The particle (or hole) excitation energy for a single particle (or hole). The branching ratio required during the closed-form solution of the master equation. The ratio between state density for non-ESM to that for ESM systems. Plank’s constant. Reduced mass of the emitted particle plus the residual nucleus. A subscript indicating neutron-type. The transition probability of the type (x) with (y)=sign(∆n). Example: λπ+ is the transition of proton type with ∆n=+2. The reduced wavelength of the projectile. A subscript indicating proton-type. The Heaviside step function of the variable x. Total Level Density. Reaction cross-section. Geometrical cross-section. The spin cut-off parameter for the precompound nucleus at exciton state n. The total spin cut-off parameter for the precompound nucleus. The inverse cross-section for the particle β, this is the same as the absorption cross-section of the compound nucleus for a particle of type β. The inverse cross-section for the particle β, at equilibrium. It is similar to σβ(ε) but needed during NT component calculations. The lifetime of the nth exciton state. State density of the system defined by the state of p, h and E, for one- or two-component. Final accessible state density of the system.
xii
Chapter One Introduction 1. 1. Introduction Nuclear reactions caused by various types of particles have been always vital to explore nuclear structures. Various information are obtained from such reactions that provide a fundamental knowledge of the most stable and essential entity of nature: the nucleus. Two practical benefits of such development: power generation and radioactive waste managements, both of which are issues of great interest nowadays. A nuclear reaction caused by an incident particle results in sharing of the incident particle’s energy with all nucleons of the nucleus. When this sharing is complete the nucleus reaches the final or last stage, the “Thermal Equilibrium State.” A nucleus in a thermal equilibrium stage is called “The Compound Nucleus, CN.” The nuclear reaction terminates when the compound nucleus rests by emitting a nucleon (proton or neutron), a cluster (α-particle, d, t, .. etc), γ-rays, or heavy ion (as in the case of fission process). Another possibility for the CN is to terminate by the compound elastic scattering. The time scale for these CN emission processes is roughly about 10-18 sec. During the first stage of the nuclear reaction, the incident particle and the nucleus may not share the energy completely. In such a case, all (or most) of the projectile energy will be given to one or few nucleons only. Usually it will cause emission of nucleons or light complex particle. This process is like knocking out a bead or couple of beads from a gathered group, and it is responsible of very fast emission. Time scale in this case is of order of~10-24 sec. In fact, knockout reaction is only one kind of a group of fast reactions; which includes stripping, scattering and pick-up reactions.These reactions are grouped as “Direct Reactions.” In between direct and CN emissions, there is an intermediate and continuous emission of particles which is called the “Preequilibrium Emission, PE.” This is a process that occurs before the compound nucleus emission, i.e., in a time and 1
energy scales that lay between direct and CN states. This type of reaction is attributed to weak-interaction when the projectile energy is shared inside the nucleus via a series of two-body collisions. This will lead to a gradual excitation of all nucleons to an energy level which is higher than Fermi level. Each excitation will create a particle-hole pair, or an exciton pair. From each excitation step there is a small but important probability of particle emission. This emission represents the PE. The whole process is terminated by CN stage, but the changes of both of the number of particles (nucleons) and energy of the nucleus will highly affect this stage. This effect was and still an issue of interest since it reveals important information about nuclear structure. In 1966, Griffin [1] proposed the exciton model, which is a semi-classical approach to explain PE. Being a simplified model, Griffin’s exciton model could explain PE with reasonable theoretical frame. Soon after that, Blann [2-5] extended the exciton model to include system properties and features with more details. These models are mostly based on parameterization of experimental results so that general formulae can be reached to describe wide variety of nuclear reactions. Later on, many quantum-mechanical approaches were proposed to explain PE phenomena [6]. However, in one way or another, all these approaches (semiclassical and quantum mechanical) are based on Griffin’s basic idea. Details about these developments are listed in the literature survey below. The aim of any of these models is to explain available data from nuclear reactions, and to predict the results of those reactions that either cannot be experimentally achieved, or too expensive or difficult to be made. An important parameter in PE theory is the state density. The importance of the state and level densities rise from its ability to describe population of nuclear excited sates. The earliest attempts had been made by assuming that different excitation states gather in equally spaced levels and was known by the Equidistant Spacing Model (ESM). ESM represents a great simplification to a real nucleus, it gives acceptable results in one hand [5], but it is still under suspect [7] on the
2
other. The comparison of experimental data with calculations made by the ESM approach were in satisfactory agreements, in general. The field of nuclear reaction has been progressively developed since Bohr declared the “Compound Nucleus Theory” [8]. According to Weisskopf [9], nuclear reactions were thought to be categorized into two types only: Direct and CN reactions. Different types of reactions as well as emissions of many types of particles were studied according to one of these classes. These reactions may include elastic and inelastic scattering, stripping, knock-out, pick-up, fission and fusion reactions. However, all the above reactions were able to explain and reproduce part of the cross-sections for all available energy range. Both classes mentioned above will give rather recognizable structure in the energy spectra (usually peaks) during emission spectra calculations. On the other hand, the continuous distribution of particle emission available almost in all types of nuclear reactions was unexplainable according to these frames. Although these models span a wide range of nuclear reaction conditions and types, no single model was able to explain all nuclear reactions for the entire range of reaction conditions (energy, angle, momentum, mass, .. etc). CN reactions are usually treated using either Breit-Wigner or Hauser-Feshbach models -see Ref.[10] for full review on CN models. At high energies, the reactions are treated by the cascaded model [11]. Before Griffin’s model, the nuclear reactions at intermediate energies were unexplainable by these models. Therefore, the exciton model actually solved a severe problem in the nuclear reactions. In fact, this model is considered as a part of a family of models that treat the problem of nuclear reactions statistically [5], called the “Preequilibrium Models”, or “Precompound Models.” The exciton statistical model [1] added a proper base to explain these continuous particle emissions and thus it offered an unforeseen chance to regenerate the entire spectrum of nucleon induced reactions appropriately. However, inclusion of the exciton model made it possible to regenerate nuclear spectra within a good accuracy for energies up to ~150 MeV. The accuracy in these calculations depend highly on the level of details accounted for in the 3
basic assumptions of the exciton model. There are many corrections that can be added to the exciton model calculations, which improve the results in sensible manners. Most of these corrections are listed and discussed in details. Since the early days of the exciton model, the idea of ESM was used with extra care at low excitation energies and/or targets with low masses [7]. This means that there are important restrictions on applying the ESM in PE calculations. This puts a considerable difficulty in extending those models that deal with PE phenomenon. To overcome such difficulty, mainly two paths are seen: a) to obtain state and level densities with more complicated calculations that are based on numerical iterations [7, 12], b) to try the non-ESM approach [13]. The process of nuclear equilibration is thought to be responsible for the precompound nuclear emissions, where the energy of the projectile is shared with the nuclear constituents via successive processes of two-body collisions. At each state of this “Preequilibrium Phase”, there will exist a small but important probability of nuclear decay from these excited states to the continuum. Such decay is observed as a continuous spectrum laying between the direct reaction (fast emission) and the evaporation (slow) emission. The residual two-body interaction was first assumed to occur between identical particles, i.e., there is only one type of particles in the nucleus. This is the “One-component System”, or “OneComponent Fermi Gas System”. The “Two-Component System” therefore will distinguish between protons and neutrons inside the nucleus. One-component system is used to explain details of the model and its basic theory in a clear manner. The excitation development, when based on the residual two-body interaction, will ensure that the number of excitons characterizing each stage in the equilibration process will be changed by ±2 or zero. Equilibration process will cause creation of particle-hole pairs, and these pairs will be the reason that carries out the basic mechanism of energy share between the constituents of the nucleus. This will lead to successive creation of excitons during the equilibration process, and therefore, each stage in the equilibration process can be specified well by the 4
exciton number n and excitation energy E. Decay may take place from some of these stages by a certain probability λ [14].
1.2. Experimental Evidences In this section, a general review is presented about some of the experimental evidences of the preequilibrium emission. Fig.(1.1-A) gives an example of the double differential cross-section spectrum of Fe(nat) for (n,n/) reaction at 30 degrees and 14.8 MeV of excitation. The peak centered at energy near 1 MeV can be well understood by means of the CN theory as being due to evaporation process after thermal equilibration. On the other hand, the spectrum component seen at the end of the spectrum with energy about 10 MeV and more is described as due to direct reaction. The continuous region in between these two energies represent the PE of this reaction. Similar observations were found for other targets, angles, excitation energies, as well as types of particles involved in the reaction –see Figs. (1.1-B to E). In all these examples the intermediate region has a slow dependence on energy and less or no dependence on the target mass and particle type. This observation suggested that the mechanism responsible of these spectra involve different reasons other than the mechanism for direct and CN reactions. In more direct and simple words, these regions describe the path of the nucleus towards thermal equilibration. 50 45
d2σ/ dεεdΩ Ω mb/str.MeV
40 35 30 25 20 15 10 5 0 0
2
4
6
8
10
12
14
16
Energy, MeV
Fig. (1.1-A). The double-differential cross-section at 30 degrees and 14.8 MeV of excitation energy for Fe(nat) with (n,n') reaction (inelastic neutron scattering) [15]. 5
10
d2σ /dε dΩ (mb/str.MeV)
9 8 7 6 5 4 3 2 1 0 0
2
4
6
8
10
Energy, MeV
Fig. (1.1-B). The double-differential cross-section at 32 degrees and 9.2 MeV of excitation energy for 27Al with (n,n') reaction (inelastic neutron scattering) [15]. 180 160
d2 σ/ dεεdΩ Ω (mb/str.MeV)
140 120 100 80 60 40 20 0 0
1
2
3
4
5
6
7
8
Energy, MeV
Fig. (1.1-C). The double-differential cross-section at 90 degrees and 9.2 MeV of excitation energy for 209Bi with (n,n') reaction (inelastic neutron scattering)[16]. 2.5
d2σ/ dεεdΩ Ω (mb/str. MeV)
2
1.5
1
0.5
0 0
5
10
15
20
25
30
35
Energy, MeV
Fig. (1.1-D). The double-differential cross-section at 105 degrees and 33 MeV of excitation energy for 54Fe with (p,p) reaction (elastic proton scattering) [17]. 6
14 30 degrees 90 degrees 120 degrees
d2σ/ dε dΩ (mb/str.MeV)
12 10 8 6 4 2 0 0
5
10
15
20
25
30
35
Energy, MeV
Fig. (1.1-E). The double-differential cross-section at various angles and 30 MeV of excitation energy for 59Co with (p,p) reaction (elastic proton scattering) [17].
Similar remarks can be deduced from other observations such as those illustrated in Figs.(1.2-A to C), where the cross-section of other sets of reactions are shown for various types of particles. Again, the intermediate regions are analyzed only by means of the PE. In these examples one also notice the important thing that, regardless the type of the target nucleus, the intermediate region almost comes up with the same values of the cross-section (about 10-15 mb for these examples which all involve proton-neutron reactions). Such observation highly recommends the suggestion that these intermediate regions are reaction-dependent, but not target-dependent. These examples of various experimental observations clear out the basic reason behind developing the exciton model and all related PE models. All these examples can not be explained by direct or CN models. 1000 ABRAMOVICH et al SCHERY et al. WARD et al.
σ (mb)
100
10
1 0
20
40
60
80
100
120
140
160
180
Energy, MeV
Fig. (1.2-A). The cross-section for 7Li(p,n)7Be reaction [18]. 7
100
σ (mb)
١
10
1 0
20
40
60
80
100
Energy, MeV
Fig. (1.2-B). The cross-section for 34S(p,n)34Cl reaction [19]. 1000 ١
σ (mb)
100
10
1 0
20
40
60
80
100
Energy, MeV
Fig. (1.2-C). The cross-section for 48Ti (p,n)48V reaction [20].
1.3 Literature Review Due to the variety of subjects involved with the theory of preequilibrium emission, numerous papers are seen in the literature that concern the present subject. Therefore, the following review of literature will be categorized into many paragraphs, mainly each one is dealing with a specified subject. These subjects include: preequilibrium models development and applications, heavy ion induced reaction, reactions involving γ-rays, state density calculation methods, Kalbach systematics, and the master equation solution methods. Hopefully, by this way the events that made the current level of development in the statistical exciton model can be traced more clearly and reliably.
8
1.3.1 The Preequilibrium Models Development and Applications The preequilibrium statistical models are a group of models that are based on statistical approach to describe the various nuclear reactions specifically at intermediate energies. These models have been developed rapidly since the first announcement of the exciton model as a semi-classical theory to explain PE. Extended from the exciton model, the theory of PE was developed in order to include more related parameters such as the transition matrix and scattering cross-section. These models deal with the intermediate states of the nuclear reactions. Instead of the highly complicated quantum-mechanical treatment for nuclear reactions, the statistical models provided a suitable solution, and the exciton model seems an ideal model among them. This group includes exciton model [1], Hybrid Model (HM), Geometry Dependent Hybrid (GDH) model [2-5] and the Monte Carlo Hybrid (MCH) model [21-23]. The family of Hybrid models [2-5, 21-23] are improved versions of the original Griffin’s model, where the basic ideas were combined with the Master Equation Model due to Harp, Miller and Berne [24-25]. Griffin’s exciton model goes back to the year 1966 [1]. He calculated the decay rate of the equilibrium distribution for CN and took the relation of the relative probability of observing neutron emission from
117
Sn(p,n)117Sb reaction.
Basic assumption was that the PE contributes to the total spectrum by a constant factor, less than a unity. In his paper, he used many simplifications where he used one-component system of undistinguishable particles, uncorrected state density formula and ignored the ∆n=–2, and 0 transitions. Yet, his results were satisfactory. About the same time, Harp, Miller and Berne [24] proposed a purely statistical model, the Harp-Miller-Berne (HMB) model, that uses the principle of detailed balance of each excitation state to describe intermediate emission. They used Runga-Kutta-Gill method to solve the resultant system of equations and applied their calculations on a typical nuclear system with A=100, assuming excitation energy 7 to 12 MeV. However, only limited experimental spectra can be 9
analyzed according to this model, but if more correction are added to this model it can give an approximately good estimation of PE spectra [5]. In 1968, Blann [2] tried to join the principles of the exciton model together with HMB model, and the Hybrid model was emerged. This was the first important extension of Griffin's model where a try was made to evaluate the transition matrix elements for particles having relative velocity and including the particle's inverse reaction cross-section, σβ(v), in these calculations. Blann suggested further the Geometry-Dependent Hybrid (GDH) model [3] where more details about the mechanism of nucleon interactions and scattering kinematics were considered. Blann also provided a series of important papers where he used more dedicated calculations for selected spectra [26-28], with realistic calculations of the state density [12]. He also wrote an excellent review of the exciton models [5] that summarized the various development of these models until 1975. Cline and Blann [26] put the exciton model under more elaborate study where full details about the master equation and transition rates were considered for onecomponent system. The form of the reaction matrix element was assumed to be energy dependent with simple formula, and the model was parameterized for the first time to include type of the incident particle. Their work was a starting point to many similar researches that involved many corrections and types of particles. Braga-Marcazzan et al. [29] applied PE calculations for mass region with A>100 and excitation energies 10-20 MeV for (p,n) reactions cross-section. Gadioli et al. [30] proposed few important corrections to the model such as charge and spin distribution effects. They applied PE for a variety of targets, namely, 89Y, 90
Zr,
160
Gd,
169
Tm,
170
Yb, and
181
Ta for the (p,n) reactions at energy range 10-50
MeV. This one-component analysis gave an important study for many effects such as exciton scattering and non-ESM effects. Gadioli et al. [31] made further and careful analysis of PE spectra for 28 reaction types including 10-100 MeV protons hitting on 88Sr, 89Y and 90Zr targets with (p,ypxn) reactions (y=0 to 4 and x=1 to 5). Again this study used one-component system and the aim was to specify further the effects of exciton scattering and charge effect on PE spectra, as well as those of 10
emission rates and evaporation components. Good matching was obtained for spectra up to (p,pn) and (p,3n) reactions; but the results were worsened for (p,2pn) and higher for (p,4n) and (p,5n) reactions. They attributed these inconsistencies to the absence of a proper charge factor. However, their charge factor was under suspicion [27], but some modified versions of the charge factor they suggested were proposed by Kalbach [14] and Dobeš and Bĕták [32]. Blann made the various models of PE with more numerical examples and showed that the inclusion of a correct value of Fermi energy is crucial to the calculations [27]. The quantum-mechanical version of the exciton model appeared in full details for the first time in 1980 due to Feshbach, Kerman and Koonin [6], the "FKK Theory". The basic assumption in this theory was that the reaction may be understood by means of two stages, namely, the multi-step direct (MSD) and multi-step compound (MSC). The MSD component assumes that the particles involved in the reaction are mostly unbounded with at least one bounded particle, and the possible emission from this stage is of relatively short time and high energy with significant forward peak. In the MSC stage, all particles are bounded and the emission is rather slow with continuous distribution, i.e., symmetric about 90 degrees. In 1981, Bonetti et al. [33] applied the FKK theory to study (p,n) reactions on
40
Ca, 90Zr,
120
Sn, and
208
Pb at energies 15 to 45 MeV. Bonetti et al.
[34] also used similar calculations to study and analyze (p,n), (n,p) and (3He,p) reactions with 25Mg, 27Al, 40Ca, 59Co, 89Y and
103
Rh at incident energies of 13, 14,
14.5, 14.7, 14.8 and 18 MeV. Both of these comprehensive studies were aiming toward evaluating and calculating proper reaction spectra from applying the quantum-mechanical FKK theory, in order to regenerate experimental data for nucleon induced reactions, and they provide practical investigation of the PE theory in comparison with various experimental data. Another improvement of the PE theory from quantum-mechanical point of view was due to Tamura et al.[35]. Their theory, also known as TUL theory, further extended FKK theory to include more effects such as the shape of the 11
nuclear potential well and the scattering kinematics. They applied their theory on nucleon induced reactions such as
27
Al(p,p/),
54
reactions for incident proton energy 62 MeV and
Fe(p,p/),
93
209
B(p,p/),
208
Pb(p,p/)
Nb(p,α) reaction with proton
energy 65 MeV. The first detailed extension of the exciton statistical model to describe twocomponent system was made by Dobeš and Bĕták [36]. Few but important parameterizations were made to account for the transition matrix element and transition rates for two-component system. They found analytical expression for the state density corrected for the final accessible number of states. Their important contribution was the new iterative method for solving the master equation. Applications in this case were focused on neutron and proton reactions with 54Fe and 103Rh at energies 29 and 39 MeV. The closed-form version of the exciton model was proposed by Kalbach in 1986 [14]. This offered many modifications to the model extending from the treatment of the master equation to more reliable dependence of the transition matrix element on the excitation energy and exciton number, beside careful analysis of the transition rates according to the method of Dobeš and Bĕták [36]. The exact dependence of the transition rates on state density was assumed taking into account the full two-body interaction inside the nuclear potential. In this paper Kalbach took practical examples of proton reactions with MeV, and and
54
169
54
Fe and
103
Rh at 33.5
Tm at energy 31.6 MeV; and neutron reactions with 56Fe at 22 MeV
Fe at 43.3 and 66.0 MeV. Also, Kalbach studied the double differential
cross-section based on the earlier systematics. A simple quantum-mechanical approach was proposed by Bychkov et al. [37]. At this stage it was important to have careful analysis for various models, and this was made by Blann and Vonach [38] where the goal was to start giving a unified input set of parameters for the various PE models. They applied their suggestions of input parameters on (n,xn) and (p,n) reactions on
202
Hg for energy range 14-90
MeV. This was the first attempt in this field and it was followed by many, such as
12
the IAEA consultant’s meeting in 1984 [39,40], Chadwick and Young [41], and Koning and Akkermans [42]. Cowley et al. [43] experimentally calculated differential cross-section for 90
Zr(p,p/) reaction for angles 24 to 145 degrees and energies 80 to 120 MeV. Their
aim was to examine PE spectra and analyze different angular spectra with theoretical expectations. Similar work was made by Stamer et al. [44] for (p,xn) reaction at 256 and 800 MeV on Li, Al and Pb targets. The important conclusion they reached is that the FKK theory can be applied for such high energies. Feshbach [45] further explained some difficulties in the original FKK theory. This work was extended carefully by Watanabe et al. [46] for FKK theory. They studied experimental (p,p/) and (p,n) reactions on 93Nb at 12 to 26 MeV. Few examples were illustrated for PE calculations with the inclusion of nuclear fragmentation [47] and cold fission yield [48]. Such studies were employed in PE calculations of nuclear reactions involving neutron scattering on 90
93
Nb and
Zr by Marcinkowski et al.[49]. However, inclusion of PE reactions in these
reactions was under debate [50, 51]. Blann declared the Monte Carlo Hybrid (MCH) model in 1996 [21]. This model is more involved with large exciton configurations and multiple PE’s, without facing the same difficulties that seen in the earlier HM and GDH models. He applied this method on 59
Co(p,x) and
51
90
Zr(p,xn) for the energy range 20-160 MeV, and on
V(p,x) at energies up to 200 MeV. Important modifications were
made by Blann and Chadwick to include the double-differential cross-section [22] and to include complex particle emission [23]. The MCH model was followed by a new attempt to extend the FKK theory by further including the Distorted Wave Born Approximation (DWBA) transitions to the continuum by Koning and Chadwick [52]. This paper contains a comprehensive review on the quantum-mechanical theories up to that date with various corrections. Many types of applications they chose, such as 90Zr(n,xn) at 14.1 and 18 MeV,
208
Pb(n,xn) at 14.1 MeV, 90Zr(p,xn) at 25, 45 and 80 MeV, 90Zr(p,xp) at
80 MeV, 27Al(p,xn) and (p,p) at 90 MeV, and (n,xn) reactions on each of 27Al, 56Fe 13
and
208
Pb at 113 MeV. Further work was made by Avrigeanu et al.[53] to test the
effects of nucleon-nucleon interaction strength of PE spectra. Avrigeanu et al. also studied the effect of various potential types during PE calculations [54]. A comprehensive calculations of different PE models for energy range 1 MeV to 5 GeV were made by Guimarães et al. [55]. They tried to compare the predictions of PE different models with both of the compound nucleus for low energies and the cascaded models for high energies, and came up with what they called “Cascaded Exciton Model”, which represents a combination between the two types of models for high energy limits, and detailed applications were made for 9 nuclei in the Fe region. The surface effect were further studied by Kalbach [56] where more parameterizations were made for this correction. The use of practical examples this time were focusing on neutron reactions on
27
Al,
28
Si,
59
Co,
209
Bi and
238
U at
energies 28 to 63 MeV. Even more studies were recently made by Kalbach [57] for (p,xn) and (n,xp) reactions at 28-63 MeV. The aim was to investigate the effect of missing final accessible states during PE model calculations. Also, more parameterization was made for the calculating the matrix element of the reaction. Numerous applications were made for 15 targets in the medium and heavy mass regions. Pompeia and Carlson [58] tried to describe PE spectra by a new semi-classical model that accounts for configuration mixing. However, their results were mainly concerned to the master equation solution and only few comparisons were made. Applications of the PE models for complex particles, heavy ions and fission reactions were also made by various groups [59-67]. 1.3.2 Reactions Involving γ-rays The first important attempt to include γ-ray emission from PE reactions was made by Reffo et al. [68]. They studied γ-ray emission from nuclear reactions induced by light ions where the aim was to reproduce (npγ) Bremsstrahlung model, then one can reach an appropriate approach for γ-ray emission simply from inverting the 14
principles of γ-ray absorption cross-section via Giant Dipole Resonance (GDR) model. Good results were obtained from applying the PE model on (n, γ) reaction at
93
Nb and
139
La at 14.1 MeV and for α+154Sm and 3He+148Sm reactions at 27
MeV. Obložinský [69] started investigating γ-ray emission from various nuclear reactions based on several models. He used the hybrid model of Blann and slightly extended it to account for γ-ray emission from proton induced reactions on Al, Au and Pb targets. The results of this work were pre-tested by Kopcky and Uhl [70] where various γ-ray cross-sections were found from GDR theory. Obložinský and Chadwick [71] extended this approach to include the FKK theory and to predict γray angular distribution. They applied their results on (n, γ) at 14 MeV on 93
Nb and
181
59
Co,
Ta. Chadwick et al. [72] made further investigation of Pauli blocking
by means of PE for the quasi-deuteron model of photoabsorption. Fair agreements were obtained with experimental data for the (γ,n) cross-section for Pb, Ta, Ce and Sn isotopes at different incident γ-ray energies up to 140 MeV. Most recently, Dashdorj et al. [73] investigated 48Ti(n, n/γ) and 48Ti(n,2nγ) reactions at energies 1 to 35 MeV using PE theory. A good agreement was found for (n,2n) reaction which was attributed to the successful application of PE theory in these reactions. 1.3.3 State and Level Density Calculations The importance of state density in preequilibrium emission calculations basically merges from the nature of the statistical model, where the state density gives a proper description of the states populated at a certain excitation energy. The early and simplest version of the state density formula was by Ericson[74]. Griffin[1] and Blann[2] used this simple formula in their calculations. Ericson’s formula did not include any correction for the energy nor for the number of particles involved in the states calculations. In 1971, Williams [75] extended this formula to correct the energy dependence by further taking Pauli blocking into account. Albrecht and Blann [12] applied numerical methods to count the correct number of states at each excitation energy. These numerical calculations were 15
performed for nuclei near closed shells where they applied their calculations for Sn region (Z=46, 48, 50, 52 with A=115) and for lead isotopes
206, 208, 210
Pb for
excitation energies up to 40 MeV. Williams et al. [76] made similar calculations for typical heavy nuclei with A=115 and Z=46, 48, 50 and 52 for energy range 0 to 20 MeV with exciton configurations (p,h): (2,1), (3,2) and (4,3). Blann [4] reported the importance of the unbounded particles in the state density calculation where he showed that these effects might be large for certain energies. On the other hand, Lee and Griffin [77] and Grimes et al. [78] discussed odd-even effects on state density and PE spectra for (p,n) reaction types, and reached a conclusion that such effects will be important for target nuclei that lay near closed shells. Since about the late 1970’s, there have been many improvements to account for various corrections in the state density formulae. Braga-Marcazzan et al.[29] added the effects of charge into these calculations. The effects of finite well depth [32, 79-81], shell effects [32, 79, 82] surface effect [79], pairing [82], spin, angular and linear momentum distributions [83-87], passive and active holes [88] and isospin [89-91] were thoroughly added to have better formula that is fundamentally based on the ESM. Fu [82] put a parameterization method to describe pairing effects on state density calculation, and the examples taken to obtain a general formulae that covered a wide range of mass targets and energies. The method suggested was widely accepted to correct the state density during PE calculation [92-94]. Specific applications in the paper of Fu were made for 41Ca, 94Nb, and 240Pu for energies up to 20 MeV. Herman et al. [95,96] carefully studied the effects of nuclear deformation on nuclear state density. They applied their combinatorial method to find state density for closed and nearly-closed shell, beside the deformed nuclei. An important conclusion they reached to is that as the nuclear shape becomes more deformed the state density results approach Williams’ formula [75], i.e., the formula that takes the correction due to Pauli blocking energy only. Applications were mainly on nucleon induced reactions on 90Zr at energy range 0 to 30 MeV. 16
Avrigeanu and Avrigeanu [92] presented the Partial Level Density (PLD) code for state density calculation during PE model calculation. This code included many of the necessary corrections required for optimum state density values and is assumed as a standard code by the IAEA for comparison with similar codes in the field of state density calculation. Level widths and level densities were experimentally found for mass region from 32 to 60 for energies 0.7 to 11.3 MeV by Abfalterer et al. [97]. These studies focused on the state density from the exciton model point of view. However, the state density is needed in many branches of nuclear physics such as in the calculations of nuclear statistical properties and particle distribution functions. Thus, this issue was a subject of interest for many detailed studies such as Shell Model approximate calculations [98], Shell Model Monte Carlo (SMMC) [99, 100], and projected SMMC [101-103]. In addition, because of its importance in nuclear reactions, the state and level density for deformed nuclei was a subject of interest to many studies [104, 105]. A detailed review of the state and level density calculation methods can be found in the review of Huizenga and Moretto [106]. On the other hand, many advances in the non-ESM were also made. These include the attempts of Bogila et al. [107] which did not assume any modification for the state density formula. Harangozo et al. [108] derived a formula that used pairing and Pauli blocking corrections for few terms only. Some quantum-mechanical treatments used these corrections to calculate the state density for a specified exciton configurations [109]. 1.3.4 Angular Distribution of PE and Kalbach Systematics Kalbach made many important contributions to the exciton model, where many parameters were modified or introduced to give the current version of this model. Most importantly, Kalbach studied the angular distribution of the emitted particles from reactions induced by light complex particles and nucleons in terms of empirical systematics [110, 111]. These systematics first accounted for the energy of the incident particle in the low and intermediate regions, and binding energy as well as the angular momentum of the emitted ones. It was assumed that the PE 17
occurs either due to statistical multi-step direct (MSD) or statistical multi-step compound (MSC) components, described thoroughly by means of regular Legendre polynomials with reduced coefficients. A series of papers followed were concerned more on the systematics of angular distribution taking into account the effect of unbound particles in the state density calculation [112], the possible energy dependence of the angular distribution [113], complex particle emission [14] and higher energy region [114]. Further study of these systematics were made by Costa et al. [115] where a suggestion was made that Pauli exclusion principle and Fermi motion are relevant to the angular distribution of the emitted particles, and they applied this study specially on neutron inelastic scattering. This work tried to rationalize Kalbach systematics [110,111] based on the refractive index of the incoming and outgoing waves, however, only few modifications to the original treatment were made. They applied their method on 93Nb(n,xn) reaction for energy range 10-60 MeV. Chadwick and Obložinský [87] successfully manage to show that Kalbach systematics can be clearly derived from theoretical bases when the linear momentum distribution is included in the state density, such that the equilibration process best describes the phase-space during nuclear reaction. Few applications were made for
90
Zr(p,n) at 45 MeV,
90
Zr(p,p/) at 80 MeV and
93
Nb(n,n/) at 26
MeV. These systematics are used in preequilibrium emission calculations [64-66, 93, 94] as well as in nuclear information banks for evaluated data [116, 117]. 1.3.5 The Master Equation The master equation in PE theory gains its importance from the ability to determine the time scale required for equilibration. Some authors give names as “Pauli-Master Equation”, “Boltzmann Balance Equation”, or “Harp-Miller-Berne Equation, HMB” [5]. The first attempt to solve this equation was made by Harp et al. [24,25] where they used various approximations to solve it primitively. However, their method of solution was followed by Blann [3]. Later, assuming one-component system, Luider [118] developed a time-dependent solution where initial and boundary 18
conditions are needed. Dobeš and Bĕták [119] suggested the more elaborated iterative method to solve the master equation, where the iteration may be continued until the desired accuracy is reached. However, they put their method as inconvenient for reactions with low preequilibrium fraction. Explicit solution of the master equation was suggested by Akkermans [120], where his method is similar to the method of Luider but for the extended and more realistic twocomponent system. In 1981, Gupta [121] also proposed the extension of the master equation to include two-component system. Chatterjee and Gupta [122] used element-by-element calculation method to solve these equations but they confused the lifetime of each state with equilibration time of the compound state. Another explicit solution due to Dobeš and Bĕták [36] tried to give standard solution of the master equation for two-component system. This method is explicitly used to calculate the lifetimes of substages for two-component. Kalbach [14] expended Griffin’s work [1] by adding exciton scattering interactions. The term of transition rates that include internal transitions (between substates with the same n) were also considered in the master equation, with a possibility of converting proton (or neutron) to neutron (or proton) particles, and same for holes. Kalbach [114] also put the different forms of transition rates responsible for these transitions. The most comprehensive method used for solving the master equation is the two-component solution of Herman, Reffo, and Costa [123]. It includes two-component Fermi gas system corrected for shell structure effects (mainly pairing). The master equation was written to include all possible transitions, i.e., those for ∆n=0 and ±2. The explicit form of the master equation was used and utilized to give a general solution. Recently, Jasim et al. [124] proposed a simple numeric scheme to solve the master equation that is based on Euler centered-scheme. Applications of their new method on neutron and proton reactions on 56Fe showed that this method is stable only at intermediate energies, while at low energies the results deviate about 15% from the standard methods. However, the simplicity of this method make it more appropriate for practical use during PE calculations. 19
1.4 Aim of the Present Work The aim of this work is to use the preequilibrium exciton model to study and analyze
54,56
Fe,
103
Rh and
209
Bi for (N,N) reactions, at energies less than 50 MeV.
The state densities, effective interaction matrix element, internal transition rates, master equation, decay rates, emission spectra and reaction cross-sections are to be calculated and discussed on the bases of the present development of the exciton theory. Various comparisons will be made with the literature in order to verify the consistency of the present calculations. This work also aims to study the state density completely, where a formula for the state density calculation with the non-ESM approach will be derived for onecomponent system, and compared with earlier work. This point represents primary achievement of the present work. Furthermore, the present work plans to study the mathematical methods required for solving the master equation in details. A new and simple approach to solve the master equation is to be suggested and explained, and compared with earlier methods.
20
Chapter Two Theory of the Exciton Model 2.1. Introduction It can be clearly understood from the literature survey (Section 1.3) that the development of the exciton model passed through many turning points. During the last forty years, the field of exciton model developed from the original (and simple) model of Griffin [1] to the present group of PE models. Generally, the goal of these developments was to describe various nuclear reactions with more accurate details providing better consistency with experiment and more reliability of the interpreted reaction spectra. The differences between calculated and measured reaction spectra thus become less. The current development gives quite acceptable results. In this chapter, the details of the exciton model theory will be given. A list of the symbols used in this research is given in the List of Symbols, page ix.
2.2. The Exciton Model 2. 2. 1. A General Description Weisskopf [9] showed that the best method to describe a nuclear reaction is to assume a series of weak two-body interactions between the incident particle and target nucleus. The process of two-body collision will excite one particle each time, and when it is terminated, the excitation process will result in a fully excited and thermally equilibrated entity, the “Compound Nucleus”, CN, which has a relatively long lifetime compared to the nuclear transit time [8], then decays into a certain channel. Thus, CN formation describes the equilibration process. CN theory did not pay attention to possible nuclear decay during equilibration, however, some experimental measurements could not be interpreted without making this assumption, i.e., during this process, nuclear decay is also possible. The preequilibrium model deals with nuclear emissions during equilibration process, or in other words, emission before attainment of statistical equilibrium.
21
Due to the same reason, we may see sometimes that the expression “Precompound Emission” is used, which gives the same meaning. Griffin [1] assumed that in order to explain PE, the precompound nucleus will exist in a state where “Excitons” are formed. An exciton refers to a particle-hole pair created where a particle is excited to a certain energy level leaving a hole behind. A proper reference of energy is important to distinguish between energy levels for particles and holes. This energy reference is the Fermi surface energy of the nucleus, F. Then, it might be given that particles are being excited above F and holes below it. The definition of Fermi energy is the energy value that lays half the way between the last filled and first unfilled energy levels. Exciton creation is a process that causes energy share between the incident projectile and the nuclear constituents. Actually, the basic principle of the exciton model is based on this process, where it describes the mechanism of nuclear excitation. This idea was inspired by Griffin and it is adequate for many reasons, such as: 1) Exciton creation is a two-body process, and the nuclear excitation is assumed to be also due to two-body reaction, as suggested by Weisskopf [9]. 2) Exciton creation means creating the pair (particle+hole), the particle being excited above F. This directly gives a proper description of the nuclear excitation. 3) Because it is a two-body process, the change in the exciton number, ∆n, is ±2, or zero; where these are interpreted as (a) ∆n= +2 means creation of another (new) exciton pair which means development of the nuclear excitation, (b) ∆n= −2 means annihilation of an exciton pair, which means lose of excitation as in inelastic compound nucleus scattering, and (c) ∆n= 0 means that there is either no change in the system conditions, or there is an internal conversion or scattering processes. All these cases do actually exist in a nuclear reaction with different probabilities. 4) Each exciton state may decay with a time rate that depends on the exciton number, n, and excitation energy, E. In nuclear reactions, this must be the case
22
because the decay rate must depend on number of excited particles (population of excited particle states) and excitation energy. 5) When many numbers of excitons are created, the strength of the exciton creation process becomes less significant. This means that as the excitation states developed and grow in number, exciton creation becomes less effective due to the reduction of the excitation energy which lowers the chances of exciton creation. 6) In Griffin’s model, emission is likely to occur from states having the most probable exciton number, n . This is also seen in nuclear reactions where there is a certain state with most probable chance to decay. 7) There is a maximum exciton number nmax which represents the final possible exciton state. In nuclear reactions, also a final excitation state must exist. 8) At nmax, no further exciton creation is possible. In terms of nuclear reactions, it is said that the nucleus reached thermal equilibrium and the CN is formed. Thus, choosing exciton idea to represent nuclear reactions is quite suitable. These are the main reasons behind the popularity of Griffin’s exciton model. 2. 2. 2. Basic Assumptions In Figs.(2.1 and 2.2), an example is shown for the first few stages of nuclear reaction in the exciton model. The development of the configurations is schematically shown in Fig.(2.1). The importance of some of the realistic corrections is obvious. For example, the correction due to Pauli principle prohibits the existence of two (or more) identical particles in the same state. Therefore, this will force the configuration of Fig.(2.1-C), for example, to be rearranged so that Pauli principle is not violated. One also expects that for closed shells the particles’ excitation energy would be higher than for particles in non-closed shells. Another expected behavior is that paired nucleons would require higher energy to be excited more than unpaired ones. Note that in Fig.(2.1) one assumed that the nucleus is treated as an entity made of one type of particles, which is called the one-component system.
23
E
incident particle
empty levels …etc occupied levels
(B) first excitation (2p, 1h)
(A) before excitation (1p, 0h)
Incident Particle
(C) second excitation (3p, 2h)
Fig.(2.1) Schematic representation of the excitation process (for one component Fermi gas system) and exciton creation during nucleon-induced reaction [10]. Fermi Surface
E
To Equilibrium
3-excitons (2p,1h)
Initial
5-excitons (3p,2h)
(A) Exciton State Development
U=E-B-εε
ε Β .. etc (to compound nucleus formation)
3-excitons (+1 unbounded)
5-excitons (+2 unbounded)
(B) Unbounded States Leading to PE
Fig.(2.2). Schematic representation of the first few stages of nuclear reaction in the preequilibrium statistical model [26]. 24
In a two-component system, the situation becomes more complicated because other restrictions are added such as the difference in the binding energies and pairing. Another restriction is seen from the figure is that the maximum number of excitons is limited to (2A+Aa). However, this is considered as a severe limit because it requires that all the nucleon particles inside the nucleus are moved to excited levels, which requires the formation of the compound nucleus. This extremely excited limit is usually not considered in the practical uses of the exciton model [1]. The basic events of the exciton model are [26]: 1- The incident particle will interact first with one nucleon inside the target nucleus and form an initial exciton configuration. This initial configuration is caused by the incident projectile. Its values are usually given as: no = (po + 1, ho +1). This is indicated in Fig.(2.1), with initial exciton is assumed to be (1p, 0h). 2- The system then passes to other exciton states by the process of transition. This is mainly caused by the interaction of the exciton configuration with other nucleons inside the nucleus, leading to ∆n= ±2, 0 –Fig.(2.1-B and C). 3- During each exciton state, there is a small probability to have a particle in the unbound single-particle state, Fig.(2.2-A), so that a nuclear decay and particle emission may occur. This is indicated in Fig.(2.2-B) and it represents the PE. 4- Each particle emission will carry away a small reduction of the excitation energy of the system; still, the system is not equilibrated. This means that even when PE occurs, other exciton states might still be created. 5- After many states are created, and perhaps after many PE’s, the system reaches to the equilibrium state and CN is finally formed. The emission from the CN is usually characterized by different energy criteria than PE. These points represent the entire steps of the preequilibrium exciton model of nuclear reactions. The transition between different states results from the residual interaction. This interaction is specified by (a) Energy conservation, and (b) Twobody in nature, with the results of ∆n= ±2, 0. Some times these transitions are treated with equal probability, an assumption usually called “an equal a priori distribution [5,26]”, however, they still depend on the population of each state. 25
Griffin’s model was based on a “phase-space” assumption [5]. This means that only the state of energy counts in the development of a nuclear reaction. In other words, only the principal quantum number is important during the calculations, while other quantum numbers such as the spin and isospin are totally ignored. The principal quantum number plays its role in the residual two-body interaction which is described by the matrix element M. Originally, M was approximately taken as a constant for each state [1-5]. Excitons are created when the projectile incident on the target nucleus excites one particle (or more), and the process cascades from the initial configuration to higher configurations. 2.2.3. Emission Rate and the Principle of Detailed Balance To find the rate of particle emission, Wβ(n,ε,U,E), for a system of exciton state n, emitting a particle of type β, with emission energy ε, leaving the initial excited nucleus of excitation energy E with residual energy U, then we need the following: 1- The branching ratio that describes the exciton creation. This ratio, in general, is defined as [26] the fractional decay probability from a given initial state to one of several final states of the system, usually given in percentage. Thus,
Branching Ratio =
Decay Probability from Initial State Decay Probability to Final States
(2.1).
Since the decay probability in the statistical model depends on the density of that state [5], then the decay probability to final states is proportional to ω(n-1, U), and the decay probability from initial states to ω(n, E), thus, ω (n − 1, U ) (2.2). ω ( n, E ) 2- The transmission frequency, which represents the average frequency of the Branching Ratio =
particle inside a potential barrier before it transits through the walls of the confining potential. This frequency is similar to the probability of α-particle tunneling [26], and it is given here as, Transmission Frequency =
σ (ε ) vo Vnucl.
3- The free-particle phase-space factor, which is given as [26], 26
(2.3),
Free - Particle Phase - Space Factor =
(2s β + 1)4π pβ dpβ V (2 π h )3
nucl .
(2.4).
Then, the rate of particle emission Wβ(n,ε,U,E), will be given as, ω (n-1, U ) σ (ε ) vo Wβ (n, ε , U , E ) dε = (n, E ) Vnucl 1ω 42 4 43 4 1424 3 branching ratio
(2s β + 1) 4π pβ dp β dε Vnucl 3 (2 π4h2 ) 44444 dp 14444 3 β
(2.5),
phase − space
transmision frequancy
Eq.(2.5) can also be derived from the principle of detailed balance as follows [26]: Let the CN be denoted by the set of states C ; where this set is specified by p, h, and E, and let R be the set of states describing the residual nucleus plus the emitted (free) particle. The set R is specified by the type of the particle, β, its energy ε, particle number (p-1), hole number h, and energy U. If the system suffers a transition from the initial state described by C to the final state R then the following relation holds, * N C WCR (n, ε , U , E ) = N R WRC ( n, ε , U , E )
(2.6),
which is called “Detailed Balance Principle”. Then, WCR (n, ε , U , E ) =
NR * WRC (n, ε , U , E ) NC
(2.7).
The numbers NC and NR are found from, NC = ω ( p, h, E) dE
N R = [ω ( p − 1, h,U ) dE ] ωβ (ε ) dε
[
]
(2.8).
Then, eq.(2.7) becomes, ω ( p − 1, h, U ) ω β (ε ) * WRC (n, ε , U , E ) dε (2.9). ω ( p , h, E ) This equation in, general, is similar to Hauser-Feshbach formula [125], but it is WCR (n, ε , U , E ) =
used for emission from preequilibrium states. During the reaction, the exciton model uses a description for the time-reversed transition rate proportional to the inverse cross-section σβ(p-1,h,ε,U), as [26], * WRC (n, ε , U , E ) =
vβ σ β ( p − 1, h, ε , U ) Vnucl.
(2.10),
The capture occurs to one state within ωβ(ε) dε states, formulated as follows [26], ω β (ε )dε =
Vnucl. (2sβ + 1)(2µ β )3 / 2 ε dε 2 3 4π h 27
( 2.11).
Thus, putting: vβ = 2ε , eq.(2.9) will be given as, µβ ∴WCR (n, ε , U , E ) =
(2sβ + 1) µ
βε σ β ( p
− 1, h, ε , U )
ω ( p − 1, h, U ) dε ω ( p, h, E )
( 2.12).
π h For complex particle emission, such as α-particle, this equation may not hold; 2
3
therefore, additional factors are needed to correct this equation. It was stated [5,7] that the cross-section σβ(p-1,h,ε,U) is difficult to evaluate, and thus it might be replaced by the capture cross-section σβ(ε) for the ground state. However, Kalbach [93,94] used systematic calculations for this cross-section based on a suggestion due to Gupta. Dostrovksy et al. [126] used a simple approach for this problem. Then, eq.(2.12) will now be given in the following form,
(2sβ + 1) µ
ω ( p − 1, h, U ) dε (2.13). ω ( p , h, E ) π h This equation is used to calculate particle emission rate due to preequilibrium ∴Wβ (n, ε , U , E )dε =
2
3
β
ε σ β (ε )
reactions. Notice that eq.(2.13) is more detailed than eq.(2.5). This equation was also derived by Griffin[1] and Blann in [2], but using different approach. Eq.(2.13) is used for one-component system. For the more corrected two-component system, a similar approach can be used to find the proton emission rate as [26], ∴Wβ (n, ε , U , E ) =
(2sβ + 1) µ π h 2
3
β
ε σ β (ε )
ω ( pπ − 1, hπ , pν , hν , U ) ω ( pπ , hπ , pν , hν , E )
(2.14).
2.3.The State Density The state density of the precompound nucleus is defined as [7]: the number of states per unit energy for a given state described by the exciton number, n, and excitation energy E. In the exciton model, the state density is an important quantity where the main calculations are based on. The state density is needed in the calculations of PE because it provides a useful description of the statistical properties of the CN[10]. This quantity is sometimes called “Partial Level Density, PLD” in order to distinguish it from the “total level density”, ρ(E). The total level density is the sum of the state density for all possible exciton number, n, at a given excitation energy, E, i.e.,
28
ρ (E) =
∑ ω(n, E)
(2.15).
n
The quantity ω(n,E) is a nuclear property that is not easy to measure experimentally [106]. This difficulty rises from the instability of the nuclear excited states. Theoretical formulations can only approach the value of the nuclear state density by approximate formulae. This is mainly due to the lack of the knowledge about the exact nuclear structure [7]. Therefore, all the theoretical calculations of the state density are limited to some conditions. It was confirmed that ω(n,E) increases rapidly with n and E, therefore the relation between ω(n,E) and E is exponential with n. Almost all experimental calculations of the total level density, ρ(E), can be fitted to an exponential form [10]. Examples are [10],
[ ] exp[2E a ] ρ (E) = C ρ (E) = C exp 2 aE
E2 exp 2 aE ρ (E) = C E
[
]
(2.16),
where C and a are fitting constants that vary accordingly. The Equidistant-Spacing Model (ESM) approximation of the nuclear state density is described by the single particle state density per each MeV labeled as g. In ESM, the energy difference between any two successive energy levels is, ∆En = En+1 – En = 1/g = constant
(2.17),
and the idea is schematically illustrated in Fig.(2.3). Although the ESM sounds like a crude approximation, it gives acceptable results in PE calculations [7,14,26]. It is usual to use the expression “Fermi Gas System” to describe the mechanism of nuclear equilibration. This expression means that the constituents of the nuclear matter are basically non interactive with each other and that the nuclear potential is constant within the nuclear volume [127].
29
Ε n+1 d= =∆Εn
Εn
(a)
(b)
Fig.(2.3). Schematic representation of (a) non-ESM and (b) ESM approximation of the energy level structure of the nucleus. Note that at intermediate energies both representations will be approximately the same. This justifies the validity of the ESM approximation for these excitation energies.
The exciton number, n, is defined as n=p + h
(2.18-a),
when it is assumed that protons and neutrons are indistinguishable particles. For a two-component Fermi gas system, we have, n = nπ + nν = pπ + hπ + pν + hν
(2.18-b).
The state density for one-component Fermi gas system will be denoted by
ω1(n,E) and for two-component by ω2(n,E). The state density calculation includes the following treatments: 1- Un-corrected formula, 2- Pauli blocking, 3- Back-shifted levels and modified Pauli terms, 4- Pairing correction, 5- Surface effect and finite well depth (FWD) of the nuclear potential, 30
6- Passive and active holes effect, 7- Charge effects, 8- Spin, angular and linear momentum distributions, 9- Isospin dependence, 10- Non-ESM approach. A brief description of each of these corrections is given below: 2.3.1 Ericson’s Formula The simplest formula used to describe ω(n,E) is to exclude all possible corrections during the calculations, and it is called Ericson Formula [74]. This formula shows that state density is given for one-component system by, ω1(n, E) =
g n E n −1 p! h!(n − 1)!
(2.19) ,
where g is the single particle level density, it corresponds to eq. (2.19) itself but with exciton number set to one, i.e., g= ω1(1,ε)= ω 1(1,0, εp)= ω1(0,1, εh)
(2.20).
In the ESM approach, g is given approximately by the relation [7,14,108], g=
3A (MeV −1 ) 2F
(2.21) ,
and this relation is usually approximated by the phenomenological approximation, g≅
A A to , which is used in most practical calculations. For two-component 13 15
system, the state density is given as, ω2 (n, E ) =
g n E n −1 pπ !hπ ! pv ! hv ! (n − 1)!
(2.22),
which is written assuming that g is the same for protons and neutrons. Since for neutrons gν it is different from that of protons gπ, therefore eq.(2.22) becomes [7], gππ gνν E n −1 n
ω2 (n, E) =
n
pπ ! hπ ! pv! hv! (n − 1)!
(2.23) ,
where, gπ ≅
Z (A − Z) N g , gv ≅ g= g A A A
(2.24) .
The total level density is given by eq.(2.15) in both cases of one- and twocomponent Fermi gas systems. 31
2.3.2. Williams’ Formula The effect of Pauli correction appears as if the magnitude of the excitation energy was lowered. This means that several states are blocked, thus their contribution in the amount of the excitation energy will decrease. A net effect will be as if E becomes [E−Ap,h(p,h,E)], where Ap,h(p,h,E) is Pauli correction term [75], ω 1(E, n) =
g n (E − Ap, h )n −1
(2.25),
p! h!(n − 1)!
and Ap,h is given as [75] Ap, h =
p( p + 1) + h(h − 3) 4g
(2.26),
Eq.(2.25) is Williams’ formula for one-component. For two-component it will be, ω 2 (n, E ) =
gπnπ g vnv ( E − Apπ , hπ , pv , hv ) n −1
(2.27) ,
pπ ! hπ ! pv ! hv ! (n − 1)!
with the correction term given in this case by the following, Ap
,h , p ,h v v
π π
p ( p + 1) + h (h − 3) π
=
π
π
4g
π
+
pv ( pv + 1) + hv (hv − 3) 4 gv
π
(2.28).
The most difficulties in eq.(2.28) is that it is not symmetric in p and h, and not corrected for energy which may add some uncertainty in the calculations. Another form of equations (2.25 and 2.27) is to include a slight correction due to energy symmetry [7, 92], that is to use the formula, ω 1(E, n) =
g n (E − Ap, h ) n −1 p!h!(n − 1)!
Θ(E − α p, h )
(2.29),
where Θ(E-αp,h) is the Heaviside step function defined as, E − α p, h ≤ 0 0 Θ(E − α p, h ) = 1
(2.30) , E − α p, h > 0
and the correction term αp,h is given as [7], α p, h =
p( p + 1) + h(h − 1) 2g
(2.31),
for one-component. For two-component it is given as, gπ π g v v ( E − Apπ , hπ , pν , hν ) n −1 n
ω 2 (n, E ) =
α pπ , hπ , pv , hv =
n
pπ ! hπ ! pv ! hv ! (n − 1)!
Θ( E − α pπ , hπ , pν , hν )
pπ ( pπ + 1) + hπ (hπ − 1) pv ( pv + 1) + hv (hv − 1) + 2 gπ 2 gv 32
(2.32) , (2.33).
2.3.3. Pairing and Modified Williams' Formula Correction due to pairing will also cause a reduction in the excitation energy E by the amount of energy, ∆, that is generated from pairing effect. So, the “Effective Excitation Energy”, Eeff, of the system will be, Eeff. = E − ∆
(2.34),
If Pauli correction is added to pairing, then the effective excitation energy will be given by the general form, Eeff .= E − f(∆)
(2.35),
where f(∆) is a correction function of energy that depends in one way or another on pairing, which is given for two cases, with and without energy back-shift. Using the ESM approximation, then ω1(n,E) of a system will be given as [92], ω1 (n, E , P ) =
(
)n −1
g n E − P (∆) − B p , h
(2.36).
p! h! (n − 1)!
During the calculations of eq.(2.36), it is better to have the Heaviside step function of the form, Θ(E-P(∆)−Bp,h). This is because at small values of energy less than~10 MeV, the results of ω1(n,E) suffers from a sudden drop. This is due to the fact that [P(∆)+Bp,h] might be larger than E at low energy, therefore the Heaviside function is added to correct this. Then the suggested form of eq.(2.36) becomes, ω1 (n, E , P) =
where
(
g n E − P(∆) − B p, h p! h! (n − 1)!
( ∆ P (∆) = g
2 2 o −∆
)n −1 Θ ( E − P(∆) − B
p,h )
)
(2.37),
(2.38).
4
and the modified Pauli energy, Bp,h, is, 2g∆ 1+ n
B p, h = A p, h
The energy gap ∆ is given as [76,89], 1.60
n ∆ = 0.996 − 1.76 ∆o nc
E Ce
2
(2.39).
− 0.68
if E ≥ E phase ( 2.40),
∆ =0 ∆o
if E < E phase
33
where nc= critical exciton number, is the most probable exciton number that lead to emission, and given as follows [82], nc = 0.792 g ∆ o
(2.41),
and the condensation energy Ce given by the relation [92], ∆2o Ce = g 4
(2.42),
and Ephase is the pairing energy for phase transition [82] defined as following, 2.17 n E phase = Ce 0.716 + 2.44 n c
if
n ≥ 0.446 nc (2.43).
E phase =
0
otherwise
The set of equations (2.40 to 2.43) are obtained from curve fitting of a large body of data [82]. For example, if the energy ∆o was equal to 2.910 MeV which results from Z=N=8, then, Ce= 2.117 MeV and nc=2.304. The set of equations above must take into account the energy range above the “Effective Threshold Energy” Uth which is the threshold value of Ephase defined above. Uth is calculated from the following [92], 2 n n U th = Ce 3.23 − 1.57 nc nc
for
n ≤ 0.446, nc (2.44),
2 n U th = Ce 1 + 0.627 nc
for
n > 0.446 nc
A typical value of Uth for ESM with constant pairing energy, i.e., ∆ not a function of E, is about Uth~3.5 MeV [92]. The energy gap for the ground state, ∆o, is obtained from curve fitting of almost all known nuclei, by an equation known by Gilbert-Cameron formula [9]. This gap energy also decreases with increasing temperature and vanishes at a critical temperature which is for the ESM, 2∆ο/3.5, [92]. Above this temperature the pairing correlation disappears and the system reverts to the uncorrected condition. This is caused by the blocking effect of the quasi particles; the levels 34
occupied by them become unavailable to pairing interaction which decreases and eventually disappears. Gilbert-Cameron formula is given by [10], ∆o=∆oπ+∆oν
(2.45),
where, ∆ oπ = 1.654 − 9.58 Z × 10 − 3 ∆ ov = 1.374 − 5.16 N × 10 − 3
in MeV
(2.46).
Improved pairing was suggested by Kalbach [14], where a modified Pauli correction is added to the state density equation that accounts for pairing. For onecomponent FGM using the improved pairing correction then [94], ω1 (n, E , AK ) =
g n (E − AK ( p, h) )n −1 p! h!(n − 1)!
(2.47),
where AK (p,h) correction is given as [14], AK ( p, h) = Eth ( p, h) −
p ( p + 1) + h(h + 1) 4g
( 2.48),
where Eth differs from Uth given in eq.(2.44) above. The value of Eth determines whether improved pairing correction is included or not. If no pairing correction is included then, p2 Eth ( p, h) = m g
(2.49 − a ),
where: pm=maximum(p,h). If pairing correction is included then, Eth ( p, h) =
(
g ∆2o − ∆2 4
)+ p
2
pm + ∆2 m g
(2.49 − b).
Another improved formula is given for AK that differs from eq.(2.48) by adding a third term as [92], p ( p + 1) + h(h + 1) ( p − 1) 2 + (h − 1) 2 AK ( p, h) = Eth ( p, h) − + 4g g G p, h
(2.50),
where the new correction function is given as: E − Eth G p, h = 12 + 4 g pm
(2.51).
2.3.4. State Density with Finite Well Depth and Back-Shifted Levels When a preequilibrium state occurs, the exciton theory assumes there are certain decay probabilities to the continuum and other states (internal transition rates) from 35
that exciton state. The nuclear potential type affects both of these transitions. Thus, the energy of the emitted particles depends on nuclear potential. If one assumes a nuclear potential with finite well depth, the emitted particles will be expected to have more discriminated energies in the continuum, corresponding to well separated (non-overlapped) states. There is an important effect that is called “Continuum Effect”, which causes decrement in the state density as the particle emission energy increases. This phenomenon occurs because there was an energy sharing between bound nucleons with the emitted particle [80]. The continuum effect may change the emission rates significantly in some cases when the energy of the emitted particles is of order of the excitation energy of the nucleus, which may be due to better overlapping probability between states of the emitted particles and the nucleons inside the nucleus. In order to correct the state density for bound-states and finite depth of the nuclear potential, the shape of the potential must be defined. This important feature was first pointed out by Blann [3,5]. Many types of potential shapes are used to achieve this, of which Woods-Saxon is the most popular [81]. For one-component, the finite well depth and bound state effects on the state density are given as [7,119], p
h
gn ω1 (n, E ) = p! h!(n − 1)! i = 0
∑∑
(−1) i + j Chj C ip ×
j =0
[E − Ap, h − iB − jF ]n −1 Θ( E − α p, h − iB − jF )
(2.52 − a),
while for two-component [92], gπnπ g vnv ω 2 (n, E ) = pπ ! hπ ! pv ! hv ! (n − 1)!
pπ
hπ
pv
hv
iπ
jπ
iv
jv
∑∑∑∑
[
(−1)iπ + jπ + iv + jv ×
C pπ Chπ C pvv Chvv E − Apπ , hπ , pv , hv − iπ Bπ − jπ Fπ − iv Bv − jv Fν i
j
π
π
i
j
(
Θ E − α pπ , hπ , pv, hv − iπ Bπ − jπ Fπ − iv Bv − jv Fν
)
] n −1 × (52 − b).
The state density with exact Pauli, improved pairing correction and finite well depth correction is given for one-component as [92], 36
gn ω1 (n, E ) = p! h!
p
h
∑ ∑
(− 1)i + j C ip Chj Θ( E − Eth − iB −
i =0
j =0
n −1
( E − Eth − iB − jF ) n −1− λ B( p, h, λ ) (n − 1 − λ )!
∑
λ =0
jF ) ×
(2.53),
where the coefficients B(p,h,λ) are given as, λ
∑ C ( p , m) C ( h, λ − m)
B ( p, h, λ ) =
(2.54),
m=0
and the constants C(p,m) are determined from the recursive relations, q
m
C ( p , m) =
1 − p C ( p − 1, m − 1) bq q! g
∑ q
C (0, m) = [
1
m=0
0
m≠0
(2.55),
(2.56).
bq are Bernoulli numbers of the order q. For two-component system: ω 2 ( n, E , P ) =
(
)
g n E − P2 (∆ ) − B pπ , hπ , pv, hv n −1 pπ ! hπ ! pv ! hv !(n − 1)!
(2.57),
where the approximation gπ=gν=g was used and gπ , gν being the single particle density for protons and neutrons, respectively [92], and P2(∆) is the twocomponent pairing energy given as, P2 (∆) = P1 (∆π ) + P1 (∆ v ) g P1 (∆π ) = π ∆2oπ − ∆2π 4 g P1 (∆ v ) = v ∆2ov − ∆2v 4
( (
(2.58),
) )
(2.59 − a), (2.59 − b).
If the approximation gπ=gν=g, is used then both eq.s (2.59) will be equal. ∆oπ and ∆ον are found as functions of nπ, Eπ and nν, Eν respectively, from eq.(2.40), where,
n Eπ = E π n n Ev = E v n
(2.60),
or, within ~2% error, by the relation, ∆2oπ = ∆2ov = 4
Up
(2.61),
g
where Up is the pairing correction for the total state densities. For two-component system, the modified Pauli energy, Bpπ,hπ,pv,hv [92], 37
2g∆ 1+ n
B pπ , hπ , p v , hv = Apπ , hπ , p v , hv
2
(2.62),
where the same values of g and ∆ for protons and neutrons are assumed for simplicity. Thus, for two-component then, gπnπ g vnν [E − AK ( pπ , hπ , pv , hv ]n −1 ω 2 (n, E ) = pπ ! hπ ! pv ! hv !(n − 1)! AK ( pπ , hπ , pv , hv ) = AK ( pπ , hπ ) + AK ( pv , hv )
(2.63), (2.64),
AK ( pπ , hπ ) and AK ( pv , hv ) are defined as AK ( p, h) given by one of the former
corrections. These corrections can be defined with the function Gp,h as [92], G pi, hi =
Ei − Eth,i 12 ni + 4 gi n pm i
(2.65),
with the subscript i stands for π or ν. For two-component these equations are [92], gπnπ g vnv ω 2 (n, E ) = pπ ! hπ ! pv ! hv !(n − 1)! C ipππ
Chjππ
n −1
C ivpv
∑
Chvjv
t n − λ −1
λ =0
pπ
hπ
pv
hv
∑ ∑∑∑
iπ = 0 jπ = 0 iv = 0
Yn, λ
(n − λ − 1)!
jv = 0
Θ(t )
where the symbols used here are,
t = E − Eth − iπ Bπ − jπ Fπ − iv Bv − jv Fv
Yn ,λ =
λ
∑ B ( p , h , m ) B ( p , h , λ − m) v
v
π
(− 1)iπ + jπ + iv + jv ×
π
(2.66), (2.67 − a), (2.67 − b),
m=0
and the coefficients B(p,h,m) are found by the same way for one-component. The energy back-shift, S, is that amount of energy that is subtracted from the effective excitation energy in order to correct ω(n,E). This correction comes from the fact that when there is nucleon-nucleon interaction, the excitation energy of the system should be considered less than the actual excitation energy because some fraction of this energy is spend as kinetic energy possessed by the interacting pair. It is considered as an important shell effect. Energy shift is therefore of great importance when dealing with pairing correction. The simplest Back-Shift Femi Gas model (BSFG) equation for one-component is [92], g n (E − P(∆ ) − B − S )n −1 ω1 (n, E , S ) = p! h! (n − 1)!
Again it is more appropriate to include the Heaviside step function so that, 38
(2.68).
g n (E − P(∆) − B − S )n −1 ω1 (n, E , S ) = Θ( E − P ( ∆ ) − B − S ) p! h! (n − 1)!
(2.69).
The BSFG equation with pairing and finite well depth correction is [92], gn ω1 (n, E ) = p! h! n −1
∑
λ =0
p
h
i =0
j =0
∑ ∑
(− 1)i + j C ip Chj Θ( E − Eth − iB − jF − S ) ×
( E − Eth − iB − jF − S ) n −1− λ B ( p, h, λ ) (n − 1 − λ )!
(2.70).
For two-components [92], ω 2 (n, E , P, S ) =
(
g n E − P2 (∆) − B pπ , hπ , p v , hv − S pπ ! hπ ! pv ! hv !(n − 1)!
)n −1 ×
Θ( E − P2 (∆) − B pπ , hπ , p v , hv − S )
(2.71).
Finally, eq.(2.66) remains the same in this case, but with, t = E − Eth − iπ Bπ − jπ Fπ − iv Bv − jv Fv − S
(2.72).
2.3.5. Surface Effect and Passive and Active Holes The finite well depth and back-shift effects beside surface effect are considered as a part of shell effects, and most of these were described in a comprehensive manner by Kalbach [56,79]. Nuclear density depends on the distance from the center of mass of the nucleus. In FGM it is assumed constant at the entire nuclear volume. This can give some accurate theoretical treatment to describe the nuclear matter. However, in reality this situation can be improved if the dependence of the nuclear matter on its size was considered. There is a strongly suggested property of the nucleus that is, at the surface, the nuclear potential becomes more shallower than in the interior [56]. This adds a considerable change on state density due to an effect called the “Surface Effect”. The effect of surface structure are thought to depend on the shape of the effective nuclear potential. The mean potential depth in the region of the first interaction between the incident particle and the nucleus was parameterized as a function of the bombarding energy by Kalbach [79]. If the incident particle transfers energy above few MeV’s, the wavelength and the mean free path of the incident particle will decrease as shown in Fig.(2.4). Therefore the first interaction is likely to be allocated within the surface region. 39
Mainly, surface effect will cause a limitation of the excitation energy that is carried by one hole.
Fig.(2.4). Variation of the mean free path of a nucleon in the nuclear matter [79].
Considering the surface effect correction then [79], ω1 (n, E , V ) = ω1 ( n, E , ∞) × f1 (n, E , V ) (2.73), and ω1 (n, E , ∞) is the state density for V=∞, given by any of the above formulae. The function f1(n,E,V) is the correction due to surface effect given by [14,79], h
f1 (n, E , V ) =
∑
j h E − jV ( h)
(−1) j j =0
E
n −1
Θ( E − jV (h) )
(2.74).
Although the exact value of V is not determined for all nuclei, but the value V=Vo=38 MeV was found to give good agreements with experimentally measured transition rates. For two-component system [79,94], ω 2 (n, E , V ) = ω2 (n, E , ∞) × f1 (n, E , V )
( 2.75),
where f1(n,E,V) is the same in both cases. State density can be corrected in the initial particle-nucleus interaction by replacing the nuclear potential depth Vo by V1 . This means that instead of putting V=Vo, we put V= V1 , where V1 , is defined as
the “average effective well depth.” The choice between V1 and Vo is: V = Vo = 38 MeV V = V1
for h > 1
(2.76). for h ≤ 0 Theoretically, V1 and its average value, V1 , may be less than Vo (=38 MeV).
However, exact calculations of V1 will also require consideration of the finite well depth correction for the first stage of the nuclear reaction. The uncertainties due to the choice of V1 or V1 are not expected to be large compared to those due to deviations of g from the ESM. This is because the ESM is rather a crude 40
approximation specially at low excitation energies. At higher energies the ESM gives more reliable results. So, in order to obtain a better estimation of the surface effect, Kalbach [79] reformulated the FGM for the non-ESM. On the other hand, passive and active holes correction can be understood from the initial steps of the nuclear reaction. For a composite system formed by particles bombardment, p is usually larger than h. The nuclear potential senses the presence of particles and holes and it must have some change in its shape. All particles must have a direct influence on the shape of the nuclear potential because of their binding energies and intrinsic fields. Holes, on the other hand, may be formed during the exciton creation but still they are not sensed by the nuclear potential. In other words, it looks like if the nuclear potential does not see the presence of some holes, thus they appear as if they were nonexistent. Such holes are called by “inactive” or “passive” holes. Such a name is given in order to distinguish these holes from the “active” holes which are counted and treated by the nuclear potential in a respectful manner i.e., just like particles. The origin of active and passive holes rises from the proper treatment of the number of particles and holes in the excited nuclei. The nucleus in its ground state will basically correspond to the exciton configuration (0p, 0h) for even-even nucleus. For odd-odd or odd-A nuclei there might be an unfilled shell that appears as an excited particle from an even-even core at rest. However for the case of exciton configuration (0p, 0h) for even-even nucleus, the incident particle can be considered as a one particle degree of freedom, i.e., (1p, 0h). It will be more practical, however, if one keeps the initial configuration of the nucleus as it is, i.e., to keep p=h always. As a matter of fact in closed-shell nuclei we must have this condition for all stages [7]. This means that for a closed-shell nucleus whenever an excited particle is formed, there will always be a hole that is left behind, thus the condition p=h is always satisfied. The concept of degree of freedom is used by Kalbach in order to distinguish between active and passive holes and particles sometimes. Thus one may refer to active exciton entities (particles or holes of neutron or proton) as degrees of freedom and they will represent effective numbers 41
during the numerical calculations. However, if the exciton entities were not degrees of freedom then they might not be considered except for some special cases as shown below. In the exciton model, only those holes that represent degrees of freedom are counted. If one is dealing with a nucleus with a structure that lays far from closedshell, the problem of active and passive holes becomes less important. This correction is also ignored if both incident and emitted particles were complex particles. Also, if there were holes fixed near Fermi surface, they will be ignored. Such holes still contribute in Pauli blocking energy thus they are counted only in Pauli correction [6]. In such cases the ground states are always assumed [57] to be p = h = 0. Pauli correction is give in this case by [57], AK p, h =
q 2 p( p + 1) + h(h + 1) − g 4g
(2.77),
where q=max(p, h). 2.3.6. Charge Effect It was suggested that PE from different exciton states can also be discriminated according to the charge nature of the emitted particle. Practically, this idea is still under great debate [7]. Braga-Marcazzan et al. [29] were the first who argued to include the charge factor in PE. They fixed its value to 2/3 for (n, p) reaction at neutron energy ~14 MeV. They labeled the effective charge factor by the symbol K, which is given as the ratio of the state densities. This ratio enters in emission rate calculations, and for neutron emission it is [29], ω ( p − 1, h,U ) ω (1,1,U ) = , one-component, (2.78-a), ω ( p , h, E ) ω (2,1, E ) ω ( pπ − 1, hπ , pv , hv ,U ) ω (1,0,0,1,U ) K= = , two-component (2.78-b). ω ( pπ , hπ , pv , hv , E ) ω (2,1,0,0, E ) + ω (1,0,1,1, E ) K=
It was mentioned that for any exciton state one must have the relation [7], Kπ(n)+ Kν(n)=2
(2.79),
Another form of the charge factor was proposed [57], labeled by R(n) such that, Rπ(n)+ Rν(n)=1
(2.80),
Yet, there is a third charge factor, Q(n), defined such that [7],
42
p
β A A Qβ ( p ) = Z N
hβ
pβ ! hβ !
( p + h )!
Rβ
(2.81),
where R is the same charge factor given by eq.(2.80) and β may be π or ν. From the above, one can see that the idea of the charge factor is blurred. Indeed, the variety of definitions for this factor makes one suspects if the principle of correction to charge type of the emitted particle is valid. 2.3.7 Spin and Angular Momentum Distribution The angular distribution is important in nuclear reactions, and it was added to the model systematically. In other words, the angular distribution of the emitted particles was not driven theoretically in the exciton model like most of the corrections described above. Kalbach [111,112] found that the angular momentum distribution of the emitted particles was in all cases having a shape closer to Legendre polynomials. In an equilibrium system, the level densities are assumed to be factorized by a Gaussian distribution function of an angular momentum J, such as [107], ρ ( E , J ) = ρ ( E ) R( J ) (2.82). The distribution function R(J) is the shape distribution of the total angular
momentum, J, given as [7,102],
R( J ) =
2J + 1 3 2 2π σ ρ
2 1 J + 2 exp − 2 2σ ρ
(2.83).
In the preequilibrium states, the state density will be given in a similar way [83], ω (n, E , J ) = ω (n, E ) Rn ( J ) (2.84), In this case the angular momentum distribution function is given as [7], 2 1 J + 2J + 1 2 Rn ( J ) = exp − 2 2π σ 3n 2σ n2
(2.85).
The shape distribution function for this case is normalized as [83,87],
∑
(2 J + 1) R( J ) ≅ 1
(2.86).
J
43
It is important to distinguish between σρ and σn, where in general, the two parameters are not equal,i.e., σρ≠σn [7]. The above set of equations were mentioned by Bloch [128], and are used for high spin values. For small values of spin J, only the dependence on (2J+1) is important i.e., the shape distribution function is not important [7,86]. The spin cut-off parameter was described by many groups. All these equations can be written in the general form, σ n2 = cn n A2 / 3 (2.87), where the values of the parameter cn (constant of n)are given in Table (2.1). Table(2.1).The different values of the parameter cn used in eq.(2.87).
cn value (dimensionless)
Reference
0.16 0.282 0.26 0.24 + 3.8 x 10-3 E
[7] [86,87] [96] [95]
Chadwick and Obložinský [69,83,84] studied state densities with linear momentum. Their earlier work treated the problem for small exciton number n, and then they extended the theory to consider any exciton number. The basic concepts of the linear momentum dependence are explained from the fact that in PE there is a forward peak observed in the angular distribution of the emitted particles [14]. In order to explain that, the emission was assumed to maintain the same direction of the incident particle. As a consequence, the linear momentum was assumed and suggested to explain this peak. This idea actually made Chadwick and Obložinský be able to give more work on other mechanisms of PE that deal with γ-ray and heavy particle emission. Therefore, theoretical treatment for adding the linear momentum is indirect. This is done by adding the linear momentum to the state density calculations as an ad hoc parameter (i.e., forced parameter that has no clear origin concepts). For one-component [83]:
44
2 mE K K F2 − − if K min < K < K1 or K 2 < K < K max 2 K r π m κ (2.88), ω1 (1,1, E , K ) = if K1 < K < K 2 2 m E K 0 otherwise A where KF is the linear momentum at Fermi surface, κ = , and the 4 2 π KF 3
momentum boundaries are defined as [83], K min = 2 m E + K F2
− KF
(2.89 − a) ,
K max = 2 m E + K F2
+ KF
(2.89 − b),
K1 = 2( K F2 − m E ) − 2 K F ε
(2.89 − c),
K 2 = 2( K F2 − m E ) + 2 K F ε
( 2.89 − d ).
For n > 2 values there are similar set of equations found in ref.[83]. In general, for high exciton number one can write for any component, r
r
ω ( p, h, E , K ) = ω ( p, h, E ) M ( p, h, E , K ) where the function M(p,h,E,K) is given as [83], r M ( p, h, E , K ) =
K2 exp − (2π )3 / 2 σ n3 2σ n2
1
(2.90),
(2.91).
The spin cut-off parameter in this case is defined as, σ n2 = n < ( K proj ) 2 > (2.92), proj where K is the average square value of the momentum projection on the direction of K. 2.3.8. Isospin Dependence In the case of nuclear reaction the isospin is considered as an important quantity because there is a certain population of different states [10]. The important question is whether isospin is conserved or mixed quantum number. This was studied by Kalbach [91] by making a comprehensive study of 299 spectra of a wide range of targets, projectiles, ejectiles and bombardment energies. It was concluded that the isospin quantum number is conserved during most PE reactions. To have more insight, Kalbach presented a detailed theory of isospin effects on PE [88-91]. In brief, to calculate the effects of the isospin
45
quantum number on state density then one should consider the effect of Esym, due to isospin number from the following formula [89], ω ( p, h, E , T = Tz ) = ω ( p, h, E + Esym (T , T − 1), T , T − 1) = ω ( p, h, E + Esym (T , T − 2), T , T − 2) = ω ( p, h, E + Esym (T , T − 3), T , T − 3) = .. etc..
(2.93),
The value of Esym is found from the semi-empirical mass formula [89],
(
E sym = E sym (T , Tz ) = 110 A−1 − 133 A−4 / 3
) (T
2
− Tz2
)
(2.94).
Eq.(2.94) includes both volume and surface symmetry energies. Finally, it is possible to add an extra correction factor for the isospin quantum number, labeled by fT(p,h,T) and for any component [6, 84], as follows, ω ( p, h, E , T ) = ω ( p, h, E ) fT ( p, h, T )
(2.95).
2.3.9. non-ESM Formula The single-particle level density, g, is the key at which the expression of the state density depends. Earlier attempts were to consider variable Fermi level F [31] or more free energy dependence on u below and above Fermi surface [29,112]. Kalbach [79] discussed this dependence in some details and the conclusion was made is that, regardless the specified type of the potential well, g is expected to vary between that of the simple square well potential to the simple harmonic oscillator. A proper definition of g determines the state density type. Consider onecomponent system then in the FGM, g is given as, g (ε ) = g o
ε
(2.96),
F
For particles and holes, this equation is, ε = F + up
(2.97),
ε = F − uh Eq.(2.97) simply means that particle (or holes) energies are above (or below)
Fermi surface by the energy up (or uh). Thus, u F
g p (u p ) = g o
1+
g h (u h ) = g o
u 1− F
The state density then can be found from the relation [108], 46
(2.98).
1 ω1 ( p, h, E ) = p! h! ∞
∫
∞
∞
∫
du1( p ) g p (u1( p ) )
0
∞
du1( h) g h (u1( h) )
0
∞ ( p) ( p) du 2 g p (u 2 )... du (pp ) g p (u (pp ) ) × 0 0
∫
∫
∞ p h (h) ( h) (h) ( h) ( p) (h) du 2 g h (u 2 )... du h g h (u h )δ E − uλ − u j (2.99), j =1 λ =1 0 0
∫
∑
∫
∑
where Dirac delta function δ can be given in its integral form as, p h 1 +∞ δ (E − uλ − uj) = dk exp i k E − uλ − u j 2π − ∞ λ =1 λ =1 j =1 j =1 p
h
∑
∑
∑
∫
∑
(2.100).
If one uses this equation, then the state density can be given as, +∞
∞ 1 ω1 ( p, h, E ) = exp (ikE )dk g p (u ) exp(− i k u ) du 2π p!h! −∞ 0
∫
∫
p ∞
h
g h (u ) exp(− i k u ) du (2.101). 0
∫
One of the main contributions of the present work to the theory of PE is that we found the complete solution for eq.(2.101) which represents a new formula for state density calculation, using the non-ESM approach. This approach was given before by some authors such as Bogila et al.[107] and Harangozo et al.[108] using approximate mathematical technique to solve eq.(2.101). The approximation followed by these groups was necessary to obtain simplified solution of this relatively complicated system. The main simplification used by Bogila et al. [107] and Harangozo et al.[108] was that they both expanded eq.(2.98) using Taylor series and applied only the first three terms into eq.(2.101) to obtain a solution that accounts for Pauli blocking energy [107] and pairing energy [108]. For example, the formula reported by Bogila et al.[107] is given as, g n E n −1 ω ( p , h, E ) = o p! h! 1 4F
s+q
1 2F
p
a
h
b
∑∑∑∑
a +b
a
( −1)
s s+q
b
q
(−1)
b+q
2
2
∏ ∏C C pj
j =1
E a+b+ s+q (n − 1 + a + b + s + q )!
i h
×
i =1
(2.102)
The difference between the present work and the methods used in [107 and 108] is that, the entire expansion of eq.(2.98) is taken in this work and applied into eq.(2.101) to obtain the exact mathematical solution of this system. The method 47
suggested here does not assume only the first three terms. For simplicity, further assumptions (such as the effects of B and Heaviside step function) are omitted. These corrections will be added later. Using Taylor-Maclaurin series expansion of eq.(2.98) about u and rearranging the terms, one can write, ∞
g p (u ) = g o
p
∑
Cm u m! F
∑
Cm u m! F
m=0 ∞
g h (u ) = g o
h
m=0
m
(2.103 − a ),
m
(2.103 − b),
where, for later convenience, the defined coefficients are given in such a way that, m = 0,
1 Cm = m +1 ( 2m − 3 ) !! (−1) 2m 1 Cmh = ( 2m − 3 ) !! − 2m p
m ≥ 1, m=0 m ≥ 1,
(2.103 − c),
which reduces to eq.(2.102) if one takes the first three terms of the expansion of eq.(2.98). This clearly proves the generality of the present method, i.e., that the earlier attempts used by the former two groups [107, 108] are actually special cases of the present general and most accurate method. Let us re-write eq.(2.101) as, 1 ω1 ( p, h, E ) = 2π p!h!
+∞
∫
exp (ikE )[P (k )] p [H (k )]h dk
(2.104).
−∞
Then, the first part of the problem is to find the integral, ∞
P (k ) = g o F
∫
exp(−i k u ) 1 +
0
u du F
(2.105),
and a similar expression for H(k). The result of the solution, after some algebra, is given as, g F P (k ) = o 2 π
∞
∑ m=0 ∞
g F H (k ) = o 2 π
∑ m=0
Cmp
(iFk )m +1 Cmh
(iFk )m +1
m = 0,1,2,...
(2.106 − a),
m = 0,1,2,...
( 2.106 − b),
where the coefficients here are redefined in a different way than eq.(2.103-c) as, 48
p 3 Cm = (−1) m +1 m − ! 2 3 h Cm = − m − ! 2
(2.107).
Grouping these terms and arranging them, one reaches to the formula, g on F n 1 ω1 ( p, h, E ) = p!h! 2 n +1π (n +1) / 2
∞
∑∏
a1 , a 2 .. j =1 a p =0
+∞
∫
−∞
p
C ap j
∞
h
∑ ∏ Cbh
b1 , b2 .. λ =1 bh = 0
λ
×
exp[i k E ]dk
(iFk )
(2.108).
n + a1+ a 2 + ...+ ap + b1+ b 2 +...+ bh
Furthermore, from making the following definition, p
N =n +
h
∑ a j + ∑ bλ
(2.109),
λ =1
j =1
and solving using Cauchy’s integral formula, then the following solution is obtained, E N −1 2 n π n / 2 p! h! F N − n ( N − 1)! ) where the mathematical multiplication operator Ξ is defined as, g on
ω ( p, h, E ) =
) Ξ≡
∞
p
∑ ∏
a1, a 2 ,..a p , j =1 b1 ,b2 ,..bh = 0
Cap j
) Ξ
(2.110),
h
∏ Cbh λ =1
λ
(2.111).
It will be shown in Chapter Three that eq.(2.110) has very important characteristics, where it can be easily shown that the formulae given before by Ericson[68], eq.(2.19); Williams[69], eq.(2.25); Bogila et al.[107] and Harangozo et al.[108] can all be reproduced from eq.(2.110) using special assumptions. 2.4. The Occupation Probability Figure (2.5) illustrates the process of exciton development for a two-component system. A lot of attention is required to understand this scheme. From that figure, each stage, N, can be further used to describe each state n. This is made by choosing N and one more other degree of freedom of the system, i.e., to choose N and either pπ, hπ, pν, or hv. Also in Fig.(2.5), the transitions from any stage to another is mainly due to the two-body interaction; therefore, only neighboring 49
stages are interchangeable. In other word, we can not have direct transition from, e.g., (1,0,0,0) to the stage specified by (3,2,2,2) directly without going through the stages in between. From this figure, all the transitions can be classified into three groups [123]:1- Inter-stage transitions: with ∆N=±1 and ∆hπ=0, ±1. They are represented by the diagonal arrows. 2- Inter-substage transitions: with ∆N=0 and ∆hπ= ±1. They are represented by the horizontal arrows. 3- Rearrangement transitions: with ∆N=0 and ∆hπ=0. They are represented by the semi-circles. (1)
N is the equilibration stage number, n is the exciton number.
n=1, N=1
1000
(2)
νν
νπ
(3) (4)
2100
νν
νπ νν
πν
n=3, N=2
1011
ππ
νπ n=5, N=3
3200
νν 4300
πν νπ
πν 5400
πν 6500
νπ
etc
νπ
νπ
πν νν
3211
πν νν
4311
νν
ππ
νπ
4322
νπ νπ
ππ
νπ νπ
ππ
νπ νπ
πν
5411
2111
3222
ππ νπ
1022
ππ
νπ n=7, N=4
2122
ππ νπ νπ 3233
νπ νπ 2133
1033
ππ νπ
νπ 1044
ππ
ππ νπ
2144
etc
n=9, N=5
νπ n=11, N=6
νπ
1055
etc
Fig.(2.5). Equilibration Process of the Excited Nucleus [123]. 1) The configuration in this scheme is (pν, hν, pπ, hπ) [123]. 2) This indicates the type of the two-body interaction. ν−ν means neutron-neutron, π−π means proton-proton and π−ν and ν−π means proton-neutron or vice versa. 3) This is the rearrangement transitions. See the text. 4) This is the decay to the continuum. Shown only for few stages to keep the figure traceable.
50
The inter-stage transition is crucial for the equilibration process, where: • • •
∆hπ= 0 means there is ν−ν interaction, ∆hπ=+1 means there is π−π or π−ν interaction, ∆hπ=−1 means there is nucleon scattering (of any type). The inter-substage transition corresponds to ∆N=0 therefore ∆n=0. However,
in two-component system, this transition is accounted because it has an important effect in the master equation due to the additional quantity hπ. The physical meaning of the inter-substage transition describes the redistribution of the excitation energy among protons and neutrons in the same stage. The interaction that causes this transition takes place between unlike types of particles and it is a considerably strong interaction [123]. Finally, rearrangement transitions describe the redistribution of the excitation energy among constituents of the same state. It will not change n nor hπ, and its effect is canceled in the master equation.
2.5. The Master Equation Referring to Fig.(2.5) again, the time rate of the change of the population probability of any substage, P(N,hπ,t) is given by the “Master Equation” [123], dP ( N , hπ , t ) = dt
[λv+π+ ( E, N − 1, hπ ) + λπ+ +π (E, N − 1, hπ )]P( N − 1, hπ − 1, t ) + [λπ+ 0v ( E , N − 1, hπ ) + λ+v v0 ( E , N − 1, hπ )]P ( N − 1, hπ , t ) + [λ−v v0 ( E , N + 1, hπ ) + λπ− 0v ( E , N + 1, hπ )]P ( N + 1, hπ , t ) + [λπ− −π ( E , N + 1, hπ + 1) + λπ− −v ( E , N + 1, hπ + 1)]P( N + 1, hπ + 1, t ) + [λ0v π+ ( E , N , hπ − 1) ]P( N , hπ − 1, t ) + [λπ0 −v ( E , N , hπ + 1) ]P( N , hπ + 1, t ) − [λπ+ 0v ( E , N , hπ ) + λv+v0 ( E , N , hπ ) + λ+v π+ ( E , N , hπ ) + λπ+ +π ( E , N , hπ ) + λ−v v0 ( E , N , hπ ) + λπ− 0v ( E , N , hπ ) + λπ− −π ( E , N , hπ ) + λπ− −v ( E , N , hπ )
]
+ λ0v π+ ( E , N , hπ ) + λπ0 −v ( E , N , hπ ) + W ( E , N , hπ ) P ( N , hπ , t )
(2.112).
Note that only two numbers are used for indexing. Eq.(2.112) simply means that: 51
Time rate of change in the probability distribution P(N, hπ,t) = Number of Events Increasing – Number of Events Decreasing the Probability Distribution the Probability Distribution P(N, hπ,t) P(N, hπ,t). (2.113).
Thus, the master equation describes a balance of the population probability during equilibration process. This process terminates by the CN formation. 2.5.1.Solution Methods There are many methods used to solve the master equation, eq.(2.112). In order to save space, only three methods of solution will be given below, two of those are found in the literature and the third is developed in the present research. 1. Explicit and Analytical Solution Methods In these methods, the aim is to try finding an analytical and exact solution. The first method was due to Luider [118] and developed by Akkermans [120], which assumes one-component system. The master equation is given in a matrix form as,
[x& ] = [A ][x ]
(2.114),
where the elements of [ x& ] are, x& =
dPi dt x = Pi i
i
A = λi → j ij
(i ≠ j )
i = 1 → n, j = 1 → n
(2.115),
and for a system that looses particles, i.e., emits particles to the continuum by emission, then the following relations must hold,
∑ Ai→ j < 0
( 2.116 − a ),
i
Ai i = −
∑ A j i − Wi
( 2.116 − b),
j (≠ i)
where Wi is the emission rate of the ith particle. According to Luider, eq.(2.114) simply solves to,
[x ] = [x o ]exp([A ]t )
(2.117),
where [x o ] is the initial population of the system. The main interest is focused on the time spent by each state to reach the equilibrium condition, i.e., on, 52
∞
[T ] = ∫ [x ]dt
(2.118),
0
or on the regular form, ∞
T (nπ , nv ) =
∫
P(nπ , nv ) dt
(2.119).
0
Note that T(nπ,nν) is a short for T(pπ,hπ,pν,hν,E). Integration of both sides of eq.(2.114) gives,
[A ][T ] = −[x o ]
(2.120),
which is valid if [x(t=∞)]=0. However, if [x(t=∞)]=[x(t)] then,
[A ][T ] = [x (t )] − [x o ]
(2.121),
and the transition matrix is, λ+ 1 0 [A ] = 0 0
(
(− τ1 )−1 λ+2
0 0 λ1− (− τ 2 )−1 λ−2
0
...
...
0
0
...
with τ n = λ+n + λ−n + Wn
...
λ+n
0 0 . . . (− τ n )−1 λ−n 0
(2.122),
)−1 is the lifetime of the state with exciton number n, which
is different from T(nπ,nν). In brief, τn is the lifetime that the state n lives before it decays and T(nπ,nν) is the time before this state reaches equilibrium, so in general T(nπ,nν) > τn. Eq.(2.121) is the analytical solution of Luider. Akkermans [120] used a similar method where the matrix [A] can be explicitly inverted. Taking qo as the initial condition then the solution this time is [120], n−2 + Tn = τ n hn (qo ) j λi τ i hi j = no i = j i 2 ∆ = ∆j = 2 n
∑
∏
(2.123).
The standard explicit solution of the master equation for two-component system is due to Dobeš and Bĕták [32]. This is widely used in PE calculations and it is similar to the methods previously mentioned. Mainly, the solution of the master equation is used to give directly the lifetime (Tn=∫Pn dt) by integration of both sides of eq.(2.112), in other words this method may “jump over” the direct 53
calculation of the occupation probability, Pn and calculate its lifetime integrals. Integrating the master equation for the time range from zero to infinity, then [32], − D(nπ , nv ) = T (nπ + 2, nv )λπ− (nπ + 2, nv ) + T (nπ , nv + 2)λv− (nπ , nv + 2) + T (nπ + 2, nv − 2)λπ0v (nπ + 2, nv − 2) + T (nπ − 2, nv + 2)λ0vπ ( nπ − 2, nv + 2) + T (nπ − 2, nv )λπ+ (nπ − 2, nv ) + T (nπ , nv − 2)λv+ (nπ , nv − 2) − T (nπ , nv ) / τ (nπ , nv )
(
(2.124).
D(nπ,nν) is the initial condition and τ ( nπ , nv ) = λπ− + λv− + λπ0v + λ0vπ + λv+ + λπ+ + W
)−1 .
The method of Dobeš and Bĕták was further developed by Kalbach [14] and Herman et al. [123]. 2. Solution by Closed Forms Solution by closed forms means seeking approximate solution from approximately iterative method. The simplest solution would be given as [119], T ( n) =
1
(2.125),
λ+n which follows if one uses Griffin’s basic approximation of neglecting all the
transitions but ∆n=+2. This assumption greatly simplifies the solution. Kalbach [14] expanded both Griffin’s work [1] and Dobeš and Bĕták[32] by adding exciton scattering interactions. The term of transition rate (λπν or λνπ) in the master equation is included, with a possibility of converting (pπ – hπ) to (hν − pν) or vice-versa where this process follows from exciton creation. In order to include this effect, one can define two quantities of the same type of the occupation probability P(nπ,nν), but for specific purpose. These quantities are [14], P1(p, pπ)= the occupation probability of finding a specific state density due to pair creation (or exciton creation) from other states with (p-1) particles, P2(p, pπ)= the total occupation probability of finding a specific state density passing through a certain configuration. Both quantities represent strength of population. These quantities are explicitly defined as [14],
54
P1 ( Aa + 1, Z a ) =
λ+vv ( Z a , N a )
+ λvv ( Z a , N a ) + λπ+ π ( Z a , N a )
(2.126),
and the recurrence relation,
P2 ( Aa + 1, Z a + 1) = 1 − P1 ( Aa + 1, Z a )
and
(2.127),
[
]
P2 ( Aa + 1, Z a ) = P1 ( Aa + 1, Z a ) + P1 ( Aa + 1, Z a + 1) Γπ v ( Aa + 1, Z a + 1) × ∞
∑ [Γvπ ( Aa + 1, Z a ) Γπ v ( Aa + 1, Z a + 1)] j
(2.128),
j =0
Γvπ ( Aa + 1, Z a ) and Γπ v ( Aa + 1, Z a ) are the branching ratios and they are given as, Γπ v ( p, pπ ) = λπ v ( p, pπ ) τ ( p, pπ )
Γvπ ( p, pπ ) = λvπ ( p, pπ ) τ ( p, pπ ) Γ Γ
v
+
π+
+ ( p, pπ ) = λvv ( p, pπ ) τ ( p, pπ ) + ( p, pπ ) = λππ ( p, pπ ) τ ( p, pπ )
(2.129),
where Kalbach [14] has ignored the transition rates λ−ππ and λ−νν. τ here slightly
(
differs from the above definitions, as: τ ( p, pπ ) = λπ+ + λϖ+ + λπν + λνπ + Wn
)−1 .
For the early states of the reaction, the branching ratios will be of considerable small values and the summation of eq.(2.128) can be approximated by setting j=0. Having these considerations, then the solution of the master equation is [14], T (nπ , nv ) = P2 ( p, pπ ) τ ( p, pπ )
(2.130).
This method is now considered as a standard one for practical calculation [94]. As listed in the literature survey, there are many other methods to solve the master equation, which will not be given here. 2.5.2. The Present Iterative Method In the present work, another approach is suggested that combine the iterative and explicit methods in one simple numerical approach [124]. The master equation for one-component is, dPn P = λ−n + 2 Pn + 2 + λ+n − 2 Pn − 2 − n dt τn
(2.131).
Using a finite difference scheme to replace time derivative, writing the equations from n=1 to the nth scheme, and defining the following factors,
55
a j ,1 = ∆t λ−n + 2 ∆t a j ,2 = 1 − τn a j ,3 = ∆t λ+n − 2
where the index j is related to n by the simple relation, j =
(2.132),
n +1 . Then, the solution 2
can be simply given as [124], Pi
n +1
3
=
∑a
j , k Pi
k =1
n
j=
n +1 = 1,2,3,... 2
(2.133).
This simple equation will converge with initial conditions, n
Pi =o0 = 1 Pin= 0 = 0
(2.134),
which is convergent as long as the factors a’s are convergent. This simple method is actually based on Euler difference scheme. One may argue that this simple method is of less accuracy than the earlier methods, however, a practical comparison shows that the difference of the occupation probability between this method and earlier methods is less than~5%. Combining this level of accuracy with the relatively simple programming effort, the method suggested here shows its importance. Beside this, the method can be easily improved to include any excitons number, n. It should be mentioned that, regardless its simplicity, this method must be carefully applied with suitable choice of the time step, ∆t, like in any ordinary differential equation when seeking numerical solution. Convergence might never be reached if this parameter were of order of τn or larger. Furthermore, this method need not to be simplified by ignoring transitions other than λ-n-2 as in Refs.[26, 32] because the simplicity of this scheme makes it programmable with ease. Detailed calculations are made in Chapter Three in order to test the correctness of this method. Few comparisons are made to explain the behavior of this system at various energies.
56
2.6. Transition Rates The transition rates of the system from one exciton state to another is mainly due to the two-body interaction process [7,14,26]. There are three types of transition rates, those are specified by the type of the change in the exciton number. Thus, transitions between adjacent states are caused by these (allowed) types of transmissions. In general, Fermi golden rule is used: λ x, y =
2π | M x , y |2 ω f h
(2.135).
For a system described by the two-component master equation there are three types of the two-body transition rates described by matrices, namely, |Mππ|2, |Mπν|2 = |Mνπ|2, and |Mνν|2; for proton-proton, neutron-proton and neutron-neutron interactions, respectively. As a rough approximation, one may assume that all these interactions are the same. In general [14], 1 | M |2 ∝ E
(2.136).
Dobeš and Bĕták gave a parameterized expressions for M as follows [32], | M πv |2 = | M vv |2 =
K AN Z E K
(2.137 − a ), (2.137 − b),
A N2 R E K | M ππ |2 = A Z2 R E
(2.137 − c),
where K=fitting parameter, R=a numerical factor that accounts for different ways of interaction between like and unlike types of particles. Its value is ~2.9–3 [32]. The density ωf is found from the state density formulae described above. The theory of final accessible states is given in Refs.[7,32]. In general, the density of final accessible states is less than the state density for the same exciton number and the same energy. This is because the system will not be able to access to all available states during transition time [7]. In other words, during single interaction between any two bodies the system may not be able to access every available state. Thus, generally speaking, ωf(nf,E) < ω (n,E).
57
Transition rates are given by many authors and slightly differ from one paper to another. However, the most recent version due to Kalbach is given here [94],
[
]
E − A pπ +1, hπ +1, pv , hv n +1 gπ2 2π × λπ = h 2n(n + 1) E − A p , h , p , h n +1 π π v v +
[
]
{gπ nπ | M ππ |2 +2nv gv | M πv |2 } f (h + 1) 2π pπ hπ { gπ (nπ − 1) | M ππ |2 +2nv g v | M πv |2 } f (h − 1) λπ− = h 2
n +1 2 2π 2 pπ hπ g v f ( h) [E − Bo ( pπ , hπ , pv , hv )] | Mπ v | × λπ v = h n E − A p , h , p , h n +1 0
[
π
π
{2[E − Bo ( pπ , hπ , pv , hv )] + n | Apπ ,hπ , p ,h v
where,
v
v
v
]
− A pπ −1, hπ −1, pv +1, hv +1
(
Bo ( pπ , hπ , pv , hv ) = max A pπ , hπ , pv , hv − A pπ −1, hπ −1, pv +1, hv +1
)
}
(2.138), ( 2.139),
(2.140),
(2.141).
The transition rates for neutrons, λν+, λν− and λνπ0, are given similarly but with the subscripts (π and ν) interchanged. As for the residual two-body matrix element of the interaction, one can assume that the matrix elements |Mππ|2 =|Mνπ|2 =|Mνν|2 are the same for the purpose of model calculations; and if needed, one may write [14], |Mνπ |2 =|Mπν |2 = 3x|Mππ |2 =3x|Mνν |2
(2.142),
which was found from a wide variety of data [94]. Denoting the average by |M|2 one can write, | M |2 =
K
[A(20.9 MeV + e )]
3
(MeV) 2
(2.143),
where: e = E/n
(2.144-a),
K = constant = 1.08 x10+6 (MeV) 5
(2.144-b).
The most recent formulae due to Kalbach [57, 94], is found from experimentally evaluated transition rates, namely, A E | M | = K 3 + 20.9 go 3 A
−3
2
58
(MeV) 2
(2.145),
where K here is a constant that has the value 900 (MeV)2 for proton-proton and 2200 (MeV)2 for neutron-neutron reaction.
2.7. The Emission Spectrum A very important quantity required from PE calculations is the energy spectrum of the emitted particle or type β, that is: Iβ(ε) = (dσ/dε)β. This is the goal of the preequilibrium statistical model. It was first Griffin’s aim to calculate this quantity at intermediate energies where other models failed. The significance of PE models all lay in the calculation of energy spectrum of emitted particles at these energies. The task of calculating the energy spectrum at intermediate energies was the motivation for the present development of the preequilibrium statistical models. Important advances have been made and in all cases the need was to better fulfill and reproduce the experimentally measured spectra. Thus, various approaches have been proposed that describe the PE spectra. In the exciton model, the energy spectrum is obtained by summing over all available emission rates, Wβ(n,ε), and weighting each term of the summation by the time-integrated probability P(n,t) at each state [26]. The energy spectrum is one of the quantities that are measured experimentally during nuclear reactions; therefore, comparing the experimental spectra with the calculated ones will indicate the validity of the present model. Thus, in the exciton model, PE spectrum Iβ(ε,t) is given as [26], I β (ε , t ) dε =
(2sβ + 1) µ π h 2
3
β
ε σ β (ε ) dε
∑ T (n, t ) n ∆n = 2
ω ( pπ − Z a , hπ , pν − N a , hν , U ) ω ( pπ , hπ , pν , hν , E )
(2.146).
This formula describes the exciton model entirely. Physically it means that the measured PE spectrum composes from individual emission rates of different exciton states, weighted by the equilibration time of each state. The sum is meant to account for exciton states that differ by 2, so that the earlier discussion of ∆n=±2 or 0 is mathematically justified. From the shape of eq.(2.146) it can be seen that this simple formula includes almost all reaction specifications, except the angular momentum distribution which is added to the 59
problem from a systematic point of view, as discussed in the next paragraph. The angular momentum distribution was not adopted in the exciton model from the basic assumptions of the model -see Section 2.2. From the set of eqs.(2.138-2.141) and (2.146), it can be seen how the state density, transition rates, inverse reaction cross-section, binding and Fermi energies, as well as the master equation solution are relevant to the PE calculations. Each improvement in calculation details was made so that eq.(2.146) better approaches experimental value, and it can be clearly understood from the above discussions of this chapter that this assignment is still in progress.
2.8.Kalbach Systematics In the theory of PE, it is important to include the Angular Momentum Distribution (AMD) of the emitted particles. Griffin’s work [1] did not include any prediction of this distribution, so did Blann [2-5], however, few attempts were made [110, 111,115,129].The most comprehensive and successful description of adding AMD was a systematic approach due to Kalbach and Mann [110], where it was based on data evaluation of large bulk of experimental data. These systematics are known as “Kalbach Systematics”, or sometimes “Kalbach-Mann Systematics”. A direct and important quantity that can be found using these systematics is the doubledifferential cross-section, d2σ/dΩ dε. The AMD was added to the exciton theory first by Mantzouranis [129]. The principle idea of adding this distribution was taken by Feshbach et al. in their FKK theory [6]. To explain the relation of AMD with FKK theory, in brief, an outline of this theory is presented here. In FKK theory there are two components of the nuclear reaction that causes PE, namely: 1- The Multi-Step Direct (MSD) component. In this component there is an unbound particle in each state of the reaction. The cross-section made by this component will have a significant angular distribution. A forward peak is expected to occur due to this component.
60
2- The Multi-Step Compound (MSC) component. In this component all particles are bound at all reaction stages and the cross-section will be symmetric about 90 degrees, i.e., there is no significant peak made from this component. Then the observed cross-section is the sum of these two components. Kalbach systematics represent a phenomenological interpretation of these components. The aim of these systematics was to reproduce experimental data from various reactions with AMD. Kalbach and Mann used fitting method of large number of various experimental data for different types of incident particles and targets for a wide range of energies. In general, fitting assumed Legendre polynomials because by this way it was possible to reproduce experimental data with few parameters and the systematic could be correlated with these coefficients for various data. The aim of these systematics is to reproduce the shape of the AMD, and not the magnitude of the cross-section itself. Generally, the form of the double-differential cross-section in terms of Kalbach systematics is given as [110] d 2σ 1 dσ = dΩ dε 4π dε
l max
bl Pl (cos θ ) tot. l = 0
∑
(2.147),
where (dσ/dε)tot. is the total differential cross-section with respect to energy and the coefficients bl are the reduced Legendre coefficients related to the usual ones al by the relation [110] a bl = l ao
where ao =
(2.148 − a),
1 dσ 4π dε
(2.148 − b).
As in the FKK theory, Kalbach systematics are based on two parts, the MSC and MSD components. It is well known that [130] the continuum emission has a smooth dependence on the angular distribution where the smooth forward peak gradually increase with the incident projectile energy. Therefore it was believed that [110, 111] the details of the reaction mechanism has no significant importance on the shape of the AMD for continuum emission. On the other hand [10], direct 61
reactions have a strong dependence on the angular distribution, therefore it was thought[26,104] that there must be a certain mechanism to go from the smooth to sharp dependence on the AMD. This was made by Kalbach[110,111] so the systematics actually represent a connection between Griffin model[1] and the FKK theory [6]. Kalbach and Mann [110] also added a third component to the MSC and MSD, where particles are bound in some reaction stages and unbound in the others, and they added this to the MSC component. The systematics are based on finding the reduced Legendre polynomial coefficients bl and it was based on the following general rule [110]: Only even order of polynomials are related to MSC component, Even and odd orders of the polynomials are related to MSD component,
(2.149),
therefore, the systematics can be given in the form [110] d 2σ = ao ( MSD) dΩ dε
l max
l max
∑ bl Pl (cosθ ) + ao (MSC) ∑ bl Pl (cosθ )
l =0
(2.150),
l=0 ∆l = 2
where obviously we have,
ao = ao (MSD) + ao (MSC)
(2.151),
and the choice between this and the former eq.(2.148) is based on the experimental data under study. Originally [110], the reduced coefficients number was chosen to be 4. These coefficients depend on: 1- Angular momentum coupling, 2- Phase space, 3- Transmission coefficients. The significance dependence of energy is mainly due to the transmission coefficients, thus one will expect that bl will depend on the emission energy, ε, similarly. Kalbach and Mann [110] proposed the following for this dependence, bl =
(2l + 1) 1 + exp[ Al (Bl − ε )]
(2.152),
62
where
Al and Bl are
free fitting parameters, described systematically as [110],
Al = K1 + K 2 [l(l +1)]m1 / 2
Bl = K 3 + K 4 [l(l +1)]m 2 / 2
(2.153),
where the K’s are free variables and m’s are integers. The forms above were chosen because of the known dependence on [l(l +1)] . The values of m are [104], m1 = +1, + 2 , + 3 m2 = −1
(2.154),
and the K’s are, K1 = 0.036 MeV −1
K 2 = 0.0039 MeV −1 K 3 = 92 MeV K 4 = −90 MeV
at m1 = +2
(2.155 − a),
at m2 = −1
(2.155 − b).
Finally, some reactions were attributed as follows [110]: • Direct nucleon transfer (for nucleons) and knock-out reactions (for complex particles) are related to MSD component, • Evaporation and secondary emission of all kinds of particles are related to MSC component.
63
Chapter Three Results and Discussions 3.1. Introduction There are many details needed during the numerical calculations of PE spectra. The aim of these calculations is to reproduce experimental data with the best parameters available in order to have a better insight of nuclear reaction. In this chapter, the various calculations made in the present work are exhibited and discussed. Fair comparisons are made with the earlier works.
3.2. State Density Results Except the charge and isospin effects, all the state density corrections listed in Chapter Two are calculated and compared with the standard methods of calculation. The aim of this extended study of the state density is to check the validity of the present results with previous works, so that the current calculations can be made with confidence. Many computer codes are written during the present study. Table(3.1) lists the system properties of the present study for state density calculation. 54
Table(3.1). Specifications of State Density Calculations, for 26 Fe 28 nucleus taken as in Ref.[92]. Emax
100 MeV
nmax Single particle density, g Back-shift energy, S Pairing correction, Up Nucleon’s binding energy, B
5 A/13= 4.1538 (MeV-1) 5 MeV 3.5 MeV Proton: 8 MeV Neuron: 10 MeV Proton:38 MeV Neutron: 40 MeV
Fermi Energy, F
Potential depth, V
38 MeV
Most of the state density results are compared with the code PLD.for [92]. This code is a standard reference for state density calculations since it provides a
64
set of complete routines that include most of the correction used. Comparisons are made in the form of percentage error, as shown for each case. 3.2.1. State density calculations with no corrections, Ericson’s Formula The one- and two-component nuclear state density calculated using Ericson’s formula (with no corrections) is shown in Fig.(3.1), based on eq.(2.19) -for onecomponent) and on eq.(2.23) -for two component-. One-component configuration is (p,h)=(3,2); and the two-component is (pπ,hπ,pν,hν)=(3,2,0,0).
Fig.(3.1). The results of the state density, as a function of excitation energy E, for one- and two-component Fermi gas system based on Ericson’s formula.
This simple figure is made here to compare one- and two-component results where it can be seen that the two-component results are less than those of the onecomponent in all the investigated energy range. This behavior is expected physically because, as the nature of calculations suggests, two-component system will have to share the energy with more entities, those due to the neutron particle and holes. The effect is apparent from the concept of g, where for one-component only one entity (indistinguishable particles or holes) will share the excitation energy. Two-component, on the other hand, has the ability to distinguish the type of the exciton particle or hole, one needs to give different population of states for each type of fermions (neutrons or protons), thus define gπ and gν. Although pν and 65
hν are set to zero, the effect is still clear. As an example one can easily prove that, letting n=5 for both cases, with configurations (3,2) for one-component and
(g )nπ (Zg / A)5 Z ω (3,2,0,0) for two-component then from eq.(2.28): 2 = π n = = ω1 g5 A (g ) ≈
5
1 . So even if pν and hν equal to zero, Ericson’s formula for two-component 38.6
will still give less value of the state density than that of one-component. Fig.(3.2) shows a comparison between different configurations for Ericson’s formula for two-component system, for n=5. The configurations (3,2,0,0) and (2,3,0,0) give the same results so they are undistinguished. The configuration (4,1,0,0) is the lowest while (2,1,1,1) is the highest. The plot for one-component for n=3 was also added for comparison, where one sees that n=3 is higher than any configuration of n=5. The case of one-component with n=5 is even higher than n=3, indicating the effect of state density reduction for the two-component than that of one-component in a clear manner.
n=3
n=5
Fig.(3.2). A comparison of the state density as a function of excitation energy E for various exciton configurations of two-component system based on Ericson’s formula.
In Figs.(3.3-A and B) the results are plotted for the condition p=h for oneand two-component systems, with the summation of each case indicated. 66
Fig.(3.3-A). State density results as a function of E with the condition p=h for onecomponent system based on Ericson’s formula.
pν=hν=0 in this case
Fig.(3.3-B). The same as Fig.(3.3-A) for two-component system based on Ericson’s formula.
pν=hν= 0 is considered in the two-component system in these figures. The behavior seen from these figures indicates that higher n configuration approaches the total
67
(level) density for both cases. This means that at high exciton configuration the number of individual states per each MeV will approach the total number of states. 3.2.2. State density calculations with Pauli correction, Williams’ Formula These results are in this case based on eqs.(2.29) and (2.32), respectively for oneand two-component; and are shown in Fig.(3.4). It is known that Ericson’s formulae for one- and two-component systems actually overestimates the state density in some manner [7], thus when adding Pauli and the finite well depth corrections the state density will be decreased. It is also seen that for one-component the results are higher than those for two-component, and this is similar to the case of Ericson’s formula. A comparison is shown in Fig.(3.5) for some of these cases. Pauli blocking energy,
Ap
π , hπ , pv , hv
for the case (pν=hν=0) of the two-component, eq.(2.28), is actually not
the same for one-component, eq.(2.26). The effect here is, again, to the singleparticle density difference, gπ and gv. If one uses: A=54, Z=26, E=100, g=A/13, then,
ω 2 26 = (0.0521418) ≈ 2 × 10 −2 in this case, which explains the point. ω1 54
Fig.(3.4). The results of the state density as a function of E for one- and two-component system based on Williams’ formula. The two-component configuration is (3,2,0,0). 68
n=3
n=5
Fig.(3.5). A comparison of the state density as a function of E for various exciton configurations of two-component system based on Williams’ formula.
A comparison of this case for different exciton configurations is shown in Fig.(3.5). The same behavior seen before in Fig.(3.4) is repeated here, for the same reasons. In Figs.(3.6-A and B) the results are plotted for the condition p=h for oneand two-component systems, with the summation of each case indicated. Similar remarks can be made from these figures and Figs.(3.3-A and B).
69
Fig.(3.6-A). State density results as a function of E with the condition p=h for onecomponent system based on Williams’ formula.
Fig.(3.6-B). The same as Fig.(3. 6-A) for two-component system based on Williams’ formula.
70
3.2.3. State density calculations for Williams’ and finite well depth corrections The results of this case are shown in Fig.(3.7-A) for one- and two-component, based on eq.(2.52-a and b) respectively. A plot with linear y-axis of the same figure is show in Fig.(3.7-B) for a better comparison. Another comparison is made not for different exciton configurations as before, but for different values of F and B. The results are shown in Fig.(3.8-A) for different values of F and in Fig.(3.8-B) for different values of B. In order to compare between the three cases mentioned above, Fig.(3.9) is plotted.
54
Fe
Fig.(3.7-A). The results of the state density as a function of E for one- and two-component system based on Williams’ formula with finite well depth.
54
Fig.(3.7-B). Same as Fig.(3.7-A) with linear y-axis. 71
Fe
From Fig.(3.8-A and B) one can respectively notice that as the value of F or B increase the state density for the same system increases. From eq.(2.52), it can be seen that as F or B energies increase the value of the state density also increases. Since Fermi energy properly describes the nuclear potential depth, then increasing F will physically lead to increase state density because the wave function may exist in more number of modes. The situation is similar to the case of larger value of B. If the binding energy was equal to zero, no value of the state density is expected. Also note that the effect of F is almost the same to excitation energies ~38 MeV, and those of B ~8 MeV. These results properly explain nuclear realistic specifications (F≈38 MeV and B≈8 MeV per nucleon).
Fig.(3.8-A). The results of the state density as a function of E for one-component system based on Williams’ formula with finite well depth for different F energies.
72
Emax=100 MeV 54 Fe n=5
Fig.(3.8-B). The results of the state density as a function of E for one-component system based on Williams’ formula with finite well depth for different B energies.
Fig.(3.9). A summary of the results of the three cases for Ericson, Williams and Williams plus finite well depth (FWD) correction, as a function of E.
73
Fig.(3.9) shows how the state density improves gradually as more corrections are added during calculations, where at E~50 MeV, the state density reduced from ~107 to ~104 MeV-1, which is expected since these corrections will minimize the effective excitation energy. This improvement becomes better as more correction are added. 3.2.4. Pairing correction with and without BSFG As shown in Fig.(3.10), the state density for one-component with all possible correction specified by modified pairing, eq.(2.37)-thin dashed line-, improved Pauli term AK from eq.(2.47 and 2.48)-thick dashed line-, modified AK eq.(2.47 and 2.50) -dots-, BSFG from eq.(2.69) -thin line-, and BSFG with modified pairing and finite well depth from eq.(2.70)-thin dotted line-. The results of -ordinaryWilliams’ formula are also added for comparison. Fig.(3.11) represents the same results for two-component system, with the corresponding set of equations. The effect of adding more corrections to the state density are quite obvious where the state density keeps decreasing with each modification added. The results of BSFG start from energy ~55 MeV, which is due to the effect of the Heaviside function. If this function was not added, however, the results below this energy are totally unstable for these configuration. This point will be illustrated fairly during the comparison of the codes written during this work with PLD program [92]. Fig.(3.12-A) shows the results of different exciton configurations, on a twocomponent system with pairing correction only. From this figure, one may suggest that for two-component system, the less difference between the pairs pπ and pν in one hand, and between hπ and hν in the other hand; will make the highest state density value, i.e., as ∆p and/or ∆h ≅ 0, ω(n,E) reaches its maximum. In Fig.(3.12B) the results of different exciton number configurations are shown, on a twocomponent system with pairing using Kalbach term. The same comparison is made for the modified pairing in Fig.(3.12-C), and BSFG in Fig.(3.12-D).
74
Fig.(3.10). The calculated state density as a function of E for one-component system with pairing correction, including all cases. Those are: modified pairing -thin dashed line -, improved Pauli term AK -thick dashed line-, modified AKm -dots-, BSFG -thin line-, and BSFG with modified pairing and finite well depth -thin dotted line-.
Fig.(3.11). The calculated state density as a function of E for two-component system with pairing correction, including all cases. Those are: modified pairing -thin dashed line -, improved Pauli term AK -thick dashed line-, modified AKm –dotted line-, BSFG -thin line-, and BSFG with modified pairing and finite well depth-thin dotted line-.
75
Fig.(3.12-A). A comparison of the results of two-component system for state density calculations as a function of E, with pairing correction, with different configurations and no back-shift correction.
Fig.(3.12-B). A comparison of the results of two-component system for state density calculations as a function of E, with pairing correction and improved Kalbach correction, with different configurations. No back-shift correction.
76
Fig.(3.12-C). A comparison of the results of two-component system for state density calculations as a function of E with improved pairing correction and different exciton configurations. No back-shift correction.
Fig.(3.12-D). A comparison of the results of two-component system for state density calculations as a function of E with improved pairing correction and different exciton configurations including back-shift correction. 77
In Figs.(3.13-A and B), the results of one- and two-component formulae with Williams’ and Kalbach pairing are shown, with the condition p=h. Figs.(3.14-A and B) give the same comparisons for Williams’ plus modified Kalbach pairing, and Figs.(3.15-A and B) for the BSFG formulae for one- and two component systems, respectively. The sum of densities is also shown in each of these six figures. A glance on any pair of these figures clearly shows that the twocomponent system results are about one order of magnitude less than onecomponent results. As stated in Chapter two, the case where p=h is used for the closed shell nuclei. In Figs.(3.10 and 3.11), the p=h+1 configurations are used because the nucleus was 54Fe. If 56Fe nucleus was used, then the configurations p=h should be used instead. Figs.(3.16-A and B) show the effects of F and, B on state density for one-component. The state density results change smoothly with the value of B where as it increases the state density increases with fixing everything else during the calculations. For the values of B=50, 40 and 30 MeV, the state density results are almost identical except at the end, where a slight reduction occurred. However at B=20 MeV and less, there will be higher changes. This indicates that state density calculations at B is more close to realistic situation and that is very sensitive to this value. Also at B=8 MeV, the result is different from the next values, B=20 MeV and B=2 MeV. If one sets B=0 the state density calculations in this case gives null. This is expected since the effective excitation energy depends on Eth as well as E, B and F. The effective excitation energy is (E-Eth-iB-jF) in this case. When B (or B + F) has a value less than (E-Eth), the argument of the Heaviside function less than zero which results in null values. This is the mathematical reason. The physical reason is, at low effective excitation energy there will be less chance for exciton creation, thus no state density. The same discussion goes for the results of changing Fermi energy, F. Finally, the situation is similar for two-component since governing formulae are generally the same. The calculated value of Eth in this case is 14.05 MeV.
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Fig.(3.13-A). State density results as a function of E with the condition p=h for one-component system based on Williams’ plus Kalbach pairing formula.
Fig.(3.13-B). State density results as a function of E with the condition p=h for two-component system based on Williams’ plus Kalbach pairing formula.
Fig.(3.14-A). State density results as a function of E with the condition p=h for one-component system based on Williams’ plus Kalbach improved pairing formula.
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Fig.(3.14-B). State density results as a function of E with the condition p=h for two-component system based on Williams’ plus Kalbach improved pairing formula. Emax=100 MeV 54 Fe
Fig.(3.15-A). State density results as a function of E with the condition p=h for one-component system based on BSFG plus pairing formula. Emax=100 MeV 54 Fe
Fig.(3.15-B). State density results as a function of E with the condition p=h for twocomponent system based on BSFG plus pairing formula.
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Emax=100 MeV 54 Fe n=5
Fig.(3.16-A). A comparison of the results of one-component for state density calculations as a function of E with improved pairing and different values of F. No back-shift correction. Emax=100 MeV 54 Fe n=5
Fig.(3.16-B). A comparison of the results of one-component for state density calculations as a function of E with improved pairing and different values of B. No back-shift correction.
As a practical comparison, different values of the back-shift energy, S, are used to find the state density for one-component BSFG, shown in Fig.(3.17). From this figure it is seen that as S increases the state density decreases. The initial limit S=0 MeV corresponds to state density calculation with pairing correction only,
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while the extreme limit S=50 MeV corresponds to back-shift energy of order of separation between nuclear states. Even in such unrealistic case the state density results still vary smoothly with excitation energy E and the major difference is that the curve starts from energy~58 MeV, which is due to the shape of the Heaviside step function. Emax=100 MeV 54 Fe n=5
Fig.(3.17). A comparison of the results of one-component BSFG for state density calculations as a function of E. Different values of back-shift energies, S, are considered.
3.2.5. Surface Effect The results of this case are based on eqs.(2.73-2.75), and are shown in Figs.(3.18 to 3.20). From Fig.(3.18) the effect of surface correction is plotted as a function of E and it can be seen that as the excitation energy increases, the surface correction function decreases. The state density decreased slightly at low energies so that the curves cannot be distinguished, therefore in Fig.(3.19) the state density results are plotted for the range 70-100 MeV only for better comparison. At high energies, the surface effect added some correction to the results. Different configurations are shown in Fig.(3.20), and it is clear that the state density decreases as h increases. The results of the state density with the condition p=h are shown in Figs.(3.21-A and B) for Ericson’s formula and Williams’ formula with surface correction, respectively. 82
Fig.(3.18). The calculated surface correction as a function of E, for different values of nuclear potential. The exciton configurations was fixed at (3,2,0,0).
In all these cases the effects of surface correction is important at high excitation energies. This is because at such energies the nucleons at the surface of the nucleus, once interacted with the incident projectile, will receive much higher energy than the nucleons lying in the interior of the nuclear volume. One important note that should be added here is that the results of Fig.(3.20) are essential in judgment of the behavior of all state density calculations. If one considers the effect of pairing correction, for example, with surface effect, it will be expected that the original results of the pairing correction will behave as in this figure depending on the values of h and Vo.
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Emax=100 MeV 54 Fe n=5
Fig.(3.19). The calculated the state density as a function of E including surface effect. The interval of energy axis (70-100 MeV) is chosen for better comparison. Emax=100 MeV 54 Fe n=5
Fig.(3.20). The calculated state density as a function of E with surface correction for onecomponent, and different exciton configurations. Vo was fixed at 38 MeV.
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Emax=100 MeV 54 Fe
Fig.(3.21-A). The calculated state density as a function of E with the condition p=h for onecomponent system based on Ericson’ formula plus surface effect.
Emax=100 MeV 54 Fe
Fig.(3.21-B). The calculated state density results as a function of E with the condition p=h for one-component system based on Williams’ formula plus surface effect.
3.2.6. Angular and linear momentum distributions 3.2.6.A: To calculate the general formula of AMD, eq.(2.87), the values of the constant cn from Table (2.1) are used, and the function for the angular momentum distribution, eq.(2.85) is calculated for each case to find the nuclear state density. The results are shown in Fig.(3.22) for one-component and Fig.(3.23) for twocomponent systems, where the basic state density is assumed to be Williams’
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formula. The shape of the angular momentum distribution is shown in Fig.(3.24), taken for a range of J values, where the constant cn is assumed the energydependent cn -Table (2.1). Generally, all these distributions vary smoothly with E, however the formula proposed by Herman et al. [95] gives the lowest among them. In order to understand the behavior of AMD function, a study of this function for the dependence on the spin J is made in Fig.(3.25), for the four values of the constant cn. As expected from the shape of AMD function, all of the curves intersected with each others at the J value equals (-0.5 ћ). For this case, the leading parameter (2J+1) will be zero for all cases regardless the shape or value of the spin cut-off parameter, thus giving this behavior.
Fig.(3.22). A comparison of the results of Williams’ one-component formula for state density calculations as a function of E with angular momentum distribution.
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Fig.(3.23). A comparison of the results of Williams’ two-component formula for state density calculations as a function of E with angular momentum distribution.
Fig.(3.24). The shape of the angular momentum distribution function as a function of E for the energy-dependent cn value, taken from Table(2.1): cn=(0.24 + 3.8 x 10-3 E) [95].
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Fig.(3.25). A comparison of the shapes of the angular momentum distribution functions, R(J), for a range of J values as a function of E.
3.2.6.B: On the other hand, the results of LMD are shown in Fig.(3.26) for oneand two-component systems, and the shape of this function is shown in Fig.(3.27) for different values of n. It can be seen that the value of the LMD function decreases rapidly with energy, and the shape is more like polynomial of high order with negative arguments. The exponential nature of the linear momentum distribution function is not pure in this case because of the leading term -eq.(2.91). Then one can easily interpret the earlier figures, Fig.(3.26), where the decrease of the state density for the entire curve is explained. The physical meaning of this behavior of the state density after taking the LMD is explained that as the energy of the incident particle increases, there will be more energy share with the nucleons inside the nucleus. This fact is well known and it is the reason behind some high-energy nuclear reactions, e.g., knock-out and scattering. This also indicates that the transferred linear momentum from the projectile to the nucleus will be less and this is shown in Fig.(3.24). On the other hand, when the incident particle’s energy is high, there will be a small chance for the creation of extremely high excited states in the nucleus. This will generate excited nuclear states with large occupation density. Therefore, when the energy
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increases, there will be some chance for the particles inside the nucleus to share the highly excited states. Pauli principle will allow this with more probabilities as the energy increases, because at high excited states there will be higher values of most of the quantum numbers, thus allowing more particles to share the same energy, i.e., the system’s degeneracy will increase.
Fig.(3.26). The effect of linear momentum distribution on Williams’ formula for state density calculations as a function of E, for one- and two-component systems.
Fig.(3.27). A comparison of the linear momentum distribution function as a function of E for different exciton numbers.
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Also a study is made for the dependence of the function M(n,E,K) on n. This is achieved by changing n and plotting the results on the same figure, and the results are shown in Fig.(3.27). From this figure we see that as n increases the dependence of the LMD function on the excitation energy decreases, reaching almost a linear relation at n=7. After this, i.e., at n=8,9,..etc (not shown) the difference in the shape of this function becomes much less. This means that when there is a chance for the generation of high exciton configurations the linear momentum distribution function will tend to be a constant of energy. This important fact indicates the less importance of the linear momentum distribution at high exciton numbers, which means that if one deals with a system with exciton number n> 6, then one can safely neglects the dependence of the linear momentum with energy, but not the value. 3.2.7. Comparisons with PLD code [92] PLD is a standard computer code that was written in FORTRAN77 and distributed by Avrigeanu and Avrigeanu [92]. The code is very important because it performs calculations for the state density with various corrections and can be considered as a reference for comparisons with the results of this work found so far. In order to perform integrated work, a complete set of codes were written in this work to calculate the different state density formulae. The aim was to have our own library of codes written with Matlab programming language, which gives more control over the input/output data and can be easily joined with other calculations such as the PE spectrum and reaction cross-sections. The details of PLD program are found in Ref. [92]. In order to make this code more flexible, the entire input statements of the code are re-written so that one can fully control the input data. The code PLD was originally written to be used as it is within other codes that deal with PE calculations, therefore most of its input commands were designed for this purpose. However, these commands were all re-written and now one may specify the type of the input data more clearly and directly. This is the only improvements that are made on PLD, and the rest of the code was untouched. 90
Below are brief and direct comparisons of two of the codes written during this work for state density calculations with PLD. Each case is considered with the same input data. By this way, one can be certain about the accuracy of the results and the reasonability of the comparison. In each case, the results of the programs are compared as in a percentage error for the entire energy range. The percentage error is found from the simple equation: percentage error (%) =
ω PLD ( p,h,E ) − ω present codes ( p,h,E ) ω PLD ( p,h,E )
× 100 %
(3.1),
and the results are plotted for each case as in below. The selected examples are, respectively: eqs.(2.29 and 2.32) as in Figs.(3.28A and B) and (3.29-A and B), and eqs.(2.52-a and b) as in Figs.(3.30-A and B). In all these figures, the percentage error was always much less than 1% which
Percentage error (%)
indicate excellent matching.
Fig.(3.28-A). A comparison with the results of PLD for one-component Williams’ formula with: p=2, h=1, g=8 MeV-1, B=8 MeV, F=38 MeV.
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Fig.(3.28-B). A comparison with the results of PLD for one-component Williams’ formula, p=1, h=1, g=8 MeV-1, F=38 MeV, B=8 MeV.
Fig.(3.29-A). A comparison with the results of PLD for two-component Williams’ formula, pπ=1, hπ=1, pν=hν=0, gπ=2 MeV-1, gν=6 MeV-1, Bπ=8 MeV, Bν=10 MeV, Fπ=38 MeV, Fν=40 MeV.
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Fig.(3.29-B). A comparison with the results of PLD for two-component Williams’ formula, pπ=2, hπ=1, pν=hν=0, gπ=2 MeV-1, gν=6 MeV-1, Bπ=8 MeV, Bν=10 MeV, Fπ=38 MeV, Fν=40 MeV.
Fig.(3.30-A). A comparison with the results of PLD for one-component with finite well depth correction formula, p=2, h=1, g=8 MeV-1, B=8 MeV, F=38 MeV.
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Fig.(3.30-B). A comparison with the results of PLD for two-component with finite well depth correction formula, pπ=2, hπ=1, pν=hν=0, gπ=2 MeV-1, gν=6 MeV-1, Bπ=8 MeV, Bν=10 MeV, Fπ=38 MeV, Fν=40 MeV.
3.3. The State Density Results for non-ESM One of the most important achievements during the present work is the suggestion of eq.(2.110) for non-ESM system. In order to check the accuracy of the present treatment, comparison of the results was made with more than one standard formula of the state density. The treatment used so far gives exact mathematical solution but for approximate physical system where the violation of the condition u(h)≤F is not taken under consideration yet. Thus the aim is to test this procedure regardless these restrictions, but with two values of F: 38 MeV and 100 MeV; to perform the calculations following the necessary condition E
