r STELLINGEN
1. De door Goeppert-Mayer en Jensen gegeven tabel van aantallen verschil- lende toestanden, N.(n,J), met impulsmoment J in de n-identieke-fermion- configuraties (j)n (j=l/2,3/2,5/2,7/2,9/2 en 11/2) bevat drie fouten. Het is bedenkelijk dat deze tabel in de meeste leerboeken voor kernfysi- 2) ca ongewijzigd wordt overgenomen , ondanks het feit dat correcties hier- op zijn gepubliceerd door Goldansky en Kazarnovsky . De N.(n,J) (J>0; J heeltallig voor even n; J halftallig voor oneven n) kunnen eenvoudig wor- den berekend als de coëfficiënten in de ontwikkeling van de partitiefunc- tie (y-1) n (l+xym) = E N.(n,J)xnyJ+l m=-j n,J 33 Deze procedure kan worden gegeneraliseerd tot het geval van meerdere schillen met al of niet identieke fermionen (bosonen), terwijl ook pari- teit en isospin op analoge wijze ir: beschouwing kunnen worden genomen. 1. M. Goeppert-Mayer, J.H.D. Jensen, "Elementary Theory of Nuclear Shell Structure", New York (1955) p.64. 2. G. Hertz, "Lehrbuch der Kernphysik", Band II, Leipzig (I960) p.556. M.A. Preston, "Physics of the Nucleus", Reading, Mass. (1965) p.164. R.R. Roy, B.P. Nigam, "Nuclear Physics", New York (1967) p.240. G. Baumg'artner, P. Schuck, "Kernmodelle", Mannheim (1968) p.29. A. deShalit, H. Feshbach, "Theoretical Nuclear Physics", Vol I, New York (1974) p.944. M.A. Preston, R.K. Bhaduri, "Structure of the Nucleus", Reading, Mass. (1975) p.235. W.F. Hornyak, "Nuclear Structure", New York (1975) p.298. 3. V.I. Goldansky, M.V. Kazarnovsky, Nucl. Phys. 13 (1959) 117. l. Bij pion-inschietenergieën in de orde van 200 MeV wordt de werkzame door- snede voor pion-nucleon-verstrooiing gedomineerd door de A(1238)-resonan- tie. Als gevolg hiervan geldt bij benadering a(n pr*ir p)/a(ir p-»ir p) = a(ir n-*rc n)/a(ir ir»ir n) = 9. Dit maakt de experimentele bepaling van de ver- houding van inelastische ir— - kern-botsingsdoorsneden tot een gevoelig in- strument voor het bepalen van nucleaire golffuncties, in het bijzonder voor niet-zelfgeconjugeerde kernen in de buurt van gesloten schillen. 1. S. Iversen et al., Phys. Rev. Lett. 40 (1978) 17. 3. De a posteriori verklaring voor de schaarste aan waargenomen primaire E2-overgangen in (n,y)~reacties, gegeven door Raman et al. , is niet juist. S. Raman, M. Mizumotc 0. Slaughter, R.L. Macklin, Phys. Rev. Lett. 40 (1978) 1306.
De invloed van reuzenresonanties op overgangswaarschijnlijkheden voor statistische electromagnetische overgangen, volgend op zware-ionen-reac- ties, kan niet op consistente wijze in rekening worden gebracht op de door Liotta en Sorensen voorgestelde wijze. i. R.J. Liotta, R.A. Sorensen, Nucl. Phys. A297 (1978) f36.
let repulsieve effect van het imaginaire deel van de optische potenti- aal, zoals dat blijkt uit semi-klassieke berekeningen aan zware-ionen- reacties , kan op eenvoudige wijze gerelateerd worden aan de absorp- tieve functie van deze imaginaire potentiaal. T. Koeling, R.A. Malfliet, Phys. Rep. 22C (1975) 181.
De ontdekking van diepgebonden gattoestanden, aangeslagen in (p,d)-reac- ; ! ties , vormt een krachtig pleidooi om bij de analyse van experimentele meetgegevens fysische vooroordelen overboord te zetten, i M. Sakai, K-I. Kubo, Nucl. Phys. A185 (1972) 217.
De volksnaam "dikschieterliere" is een document voor de aanwezigheid van de grauwe gors in de provincie Groningen. Van de door Ter Laan genoem- de mogelijke vertalingen: ' tapuit (oenanthe oenanthe), 2. geelgors (eraberiza citrinella), 3. grauwe gors (emberiza calandra), is de laatste eenduidig de juiste identificatie. I. K. ter Laan, Nieuw Groninger Woordenboek, Groningen, Den Kaag (1929).
8. Gezien in het licht van de situatie op de arbeidsmarkt voor fysici ver- dient het aanbeveling het vak electronica op te nemen in het studiepakket voor studenten in de theoretische natuurkunde. 9. De voordelen van het gebruik van electronische rekenapparatuur op middelbare scholen wegen niet op tegen de nadelen daarvan.
10. tfet door Hubbeling in termen van de klassieke modale logica gegeven formele godsbewijs is mutatis mutandis ook op te vatten als duivelsbe- wijs. 1. H.G. Hubbeling, Nederlands Theologisch Tijdschrift ?£ (1972) 183.
Stellingen, behorende bij het proef- schrift van T. Koeling, augustus 1978.
il RIJKSUNIVERSITEIT TE GRONINGEN
?"i
STUDY OF THE GAMMA DEEXCITATION PROCESS OF HIGHLY EXCITED NUCLEAR STATES
PROEFSCHRIFT
TER VERKRIJGING VAN HET DOCTORAAT IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE RIJKSUNIVERSITEIT TE GRONINGEN OP GEZAG VAN DE RECTOR MAGNIFICUS DR. J.BORGMAN IN HET OPENBAAR TE VERDEDIGEN OP VRUDAG 8 SEPTEMBER 1978 DES NAMIDDAGS TE 4.00 UUR
DOOR
THIJS KOELING
geboren te Dwingeloo
Druk: VEENSTRA-VISSER OFFSET - GRONINGEN Oude Kijk in 't Jatstraat 69 f :c
PROMOTOR: PROF.DR.F.IACHELLO
COREFERENT: DR.A.E.L.DIEPERINK Aan Anneleen This work was performed as part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM), which is financially supported by the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO). CONTENTS
CHAPTER I INTRODUCTION 1
CHAPTER II MASTER EQUATION SYSTEMS 4 2.1. Introduction. 4 2.2. Properties of the solution to the homogeneous Mas- ter Equation. 7 2.3. Application to (n,Y)" and (H.I.,xttY)-reactions. 11
CHAPTER III CALCULATION OF NEUTRON CAPTURE GAMMA-RAY SPECTRA 19 3.!. Survey of previous work. 19 3.2. Outline of the model. 21 3.3. The statistical cascade. 25 3.3.1. Level density formula. 25 3.3.2. Transitions within the level continuum. 29 3.3.3. Transitions from the level continuum to the low-lying states. 32 3.3.4. Results. 39 3.4. The discrete gamma lines. 42 3.5. Multiplicities and multiplicity distributions. 43 3.6. Summary and conclusions. 49
CHAPTER IV GAMMA-RAY SPECTRA FOLLOWING (H.I.,xn)-REACTIONS 52 4.1. Introduction. 52 4.2. The model. 57 4.3. Results. 60 4.4. Discussion of the results. 67
APPENDIX A THE INTERACTING BOSON MODEL 70 A.I. Introduction. 70 A.2. Calculated spectra for 150Sm, 158Dy and 160Dy. 73 A.3. Electromagnetic transition operators in the Interac- ting Boson Model. 78 APPENDIX B THE COMPUTER CODE 80
REFERENCES 85
ACKNOWLEDGEMENTS 90
SAMENVATTING 92 CHAPTER I
INTRODUCTION
The purpose of this thesis is to provide a description of the gamma-ray deexcitation process of highly excited nuclear states, formed in nuclear reactions. Because different reactions differ on- ly in the way in which the highly excited states are initially po- pulated, the calculations presented here can generally be applied to any reaction, provided that the initial population is properly cho- sen. The new feature of this calculation is the explicit separation of the total density of states in two pieces: a discrete part which is treated explicitly and a continuum part which is treated statis- tically. In this statistical part another new and important feature is that, in addition to classifying the states according to exci- tation energy (E), spin (J) and parity (ir), the number of unpaired neutrons (n) and protons (z) is explicitly introduced. When calcu- lating electromagnetic transitions between states we thus have three kinds of transitions: (i) continuum-continuum; (ii) conti- nuum-discrete and (iii) discrete-discrete. The presence of a se- lection rule in the electromagnetic transitions between states in the continuum, introduced by the quantum numbers n and zf plays a very important and new role in determining the properties of the deexcitation process. This selection rule reflects the one-body nature of the electromagnetic transition operator and states that the quantum numbers n and z cannot change by more than two units in any given transition. Still another important difference be- tween this calculation and others is the treatment of the discrete levels, which is done here using a model which, in addition to pro- viding a good description of the known discrete states, also allows to take into account, to a good approximation, all other (experi- mentally unknown) collective states. Because these have large ma- trix elements to the low-lying states, they play an important role in the deexcitation process. In this thesis two different reactions have been studied, viz. low energy neutron capture reactions and (H.I.,xnY)-reactions. As mentioned above, the main difference between these two processes is in the initial population of the states. In fact, in the former case the initial population, representing the neutron capture sta- te, has a sharply determined excitation energy, spin and parity, whereas in the case of (H.I.,xny)~reactions we are dealing with a population distribution, both in spin and excitation energy, re- presenting the socalled entry-states, remaining after the neutron emission has taken place. We begin in Chapter II by showing that the deexcitation pro- cess of a set of excited nuclear levels by means of gamma-ray emis- sion is equivalent to a master equation system. After a more gene- ral introduction to master equation systems a master equation tech- nique for the calculation of gamma-ray spectra and the analysis of feeding times in nuclear reactions will be presented. This techni- que, combined with a cascade technique, will then be applied to (n,y)- and (H.I.,xny)-reactions in the remainder of this thesis. In Chapter III neutron capture reactions will be described in detail. A survey of previous work in this field is given in sec- tion 3.1. Details of the present method will be discussed extensi- vely thereafter. In particular, the level density formula, used for the description of the high-lying level continuum, the initial population, representing the capture state, and electromagnetic transition probabilities will be given comprehensive attention. The influence of the giant dipole resonance on El-transition probabili- ties is discussed in subjection 3.3.3. Results of calculations for 149 the case of thermal neutron capture on Sm are compared Co expe- rimental data. It will be shown that an experimental determination of multiplicity distributions of gamma-rays in coincidence with certain discrete transitions can be used as a means of extracting valuable information on the multipolarities of gamma-rays in the statistical gamma cascade deexciting the capture state. In Chapter IV (H.I.,xny)-reactions will be discussed. The procedure will then be applied to the reactions Nd(a,4n) Sm, 158Gd(a,4n)158Dy and 16OGd(ct,4n)16°Dy and the results of the cal- culations will be compared to experimental data, obtained at the KVI by Ockels et al. (Oc78). Since for the description of the low-lying (collective) states and the electromagnetic transition probabilities between them, we make use of the Interacting Boson Approximation (IBA-model) of Ari- ma and Iachello (Ar75,Ar76,Ar78a,b), in Appendix A we give a brief outline of this model and of the calculated level spectra for the nuclei 150Sm, 158Dy and 160Dy. Finally Appendix B briefly describes the organisation and flow-charting of the various computer programs, involved in the calculations. t"Ü CHAPTER II MASTER EQUATION SYSTEMS
2.1. INTRODUCTION
In nonequilibrium physics one frequently encounters a situ- ation of a system, S, of states in contact with several different reservoir systems for e.g. thermal energy, volume, matter, elec- tric charge etc. Familiar examples are: a rod of heat conducting material with different fixed temperatures at its ends, a perme- able or semipermeable membrane between two different electrolyte solutions, a system of coupled chemical reactions, where some of the reactants are continuously fed into the system and others are continuously withdrawn from it, or an electrical circuit. The time-evolution of the system S is governed by a set of coupled linear differential equations, the (inhomogeneous) roaster equa- tions,
•• ..] (2.1)
Here the p.(t) represent the occupation probabilities or the po- pulations of the states, which are labelled 1,2,...,N, at time t. The time-independent quantities T..>0 are the transition probabi- lities for going from state i to state j. The first term appearing under the summation in (2.1) represents the feeding rate of state i, due to the decay of all other states j, whereas the second term under the summation stands for the decay rate of state i into all other states j. The inhomogeneous term b. denotes the continuous feeding from or withdrawal into the reservoir systems. Master equation techniques have been applied in many fields of physics, chemistry, biology etc. For a recent review of these techniques the reader is referred to the paper by Schnakenberg (Sc76), where also explicit examples of master equation systems are given. In nuclear physics master equation techniques have been applied e.g. by Luider (Lu77,Lu78) to pre-equilibrium theory and by Bunakov (Bu77) for the description of nuclear reactions. The set of equations (2.1) may conveniently be written in vectorial notation as
|t P -r T T Nl 1 21 31 T -r T N2 12 2 32 (2.3) T T T IN 2N 3N r. denotes the total decay probability of level i, i.e. N I". Z T.. (2.4) 1 In principle the number of states, N, need not be finite and in general the limit N-*» can be performed without difficulties. The inhomogeneous master equation (2.2) can be cast into a homogeneous one by using the substitution q(t) = Ap(t) + t = ^ p(t) (2.5) The differential equation for q(t) becomes J£ q(t) = A £- p(t) = A [Ap(t)+S ] = Aq(t) (2.6) The solution of the inhomogeneous master equation (2.2) in terms of the solution of the homogeneous equation (2.6) with initial conditions qQ = q(t=tQ) = Ap(t=t0) + b = Ap0 + b (2.7) is then given by p(t) = pn + f q(s)ds (2.8) The system governed by (2.6) can schematically be represen- ted by a diagram like the one shown in fig 2.1. KVI 1342 J ,--' Fig. 2.1: Diagrammatic representation of a master equation system. The dots denote zhe states and the various possible transitions are indiaazcd by zhe arrous. The states are represented by the dots, whereas the arrows indi- cate the various possible transitions between the states. In section 2.2 we shall review some of the properties of the solution of the homogeneous master equation (2.6) and in section 2.3 the application of master equations for the calculation of gamma spectra in (n,y) and (H.I.,xnY)-reactions will be elucida- ted. 2.2. PROPERTIES OF THE SOLUTION TO THE HOMOGENEOUS MASTER EQUA- TION From the theory of systems of ordinary differential equa- tions with constant coefficients (see e.g. Ma64) it is known that for every set of initial conditions p. there exists a unique so- lution p(t) to the homogeneous master equation ^ p(t) = Ap*(t) (2.9) Ths sum of the populations of the states is a constant of the mo- I'. tion. This can be readily verified by noting that N A. . = T.. - 6. . Z T., (2.10) iJ JJ- 1J k=1 i from which it follows that N Z A.. - 0 (j=l N) (2.11) i*I J and hence N N N E p.(t)l - Z l I A..]p.(t)-0 (2.12) This conservation of total population enables us to normalize ?(t) to unity: Z p.(t) = 1 (2.13) r*l For a solution p(t) to (2.9), to be physically acceptable, we have to require (i-1, (2.14) It can be shown (Sc76) that, if the initial populations p.(t=tQ) are in the physical region, defined by (2.13) and (2.14), the so- lution p(t) belonging to this initial condition stays in the phy- sical region at any time t. For the equation (2.9) there exists at least one steady state solution p (t), i.e. a solution that satisfies 4»p (£) = 5 (2.15) In order to obtain p (t) = p , the following set of linear equa- s s tions has to be solved Ap - 0 (2.16) s The existence of a nontrivial solution to (2.26) follows from the fact that det A (2.17) which follows immediately from (2.11). It is not a priori clear that there can be found a solution to (2.16) in the physical re- gion. However, using graph theoretical methods such a solution can be constructed explicitly (the Kirchhoff solution). Furthermore, with the assumption that if there exists a transition from some state i to some state j, then the reverse transition is also possible, i.e. T. . (2.18) it can be shown, that the solution to (2.9), with the initial po- pulations in the physical region, is asymptotically stable with respect to any steady state solution in the physical region, in the sense that lim p(t) *= p (2.19) This theorem has the important consequence, that, since a solution cannot be asymptotically stable with respect to diffe- rent steady states, the Kirchhoff solution is unique. Formally the solution to (2.9) can be written as p(t) - exp[At]p0 (2.20) where exp[ At] is to be interpreted as the series expansion explAt] « I jyAV C2.21.a) A0 - fl^NxN unity-matrix) (2.21 .b) In practical applications this solution is not useful. From the theory of systems of ordinary differential equations we know that there are exactly N linearly independent solutions to (2.9). Let us denote them by p.(t),....,pN(t). If we group these solutions together to form a (fundamental) matrix Q(t) pN(t)] (2.22) we observe that this matrix satisfies (2.9), i.e. 10 5£ Q(t) = (2.23) Moreover this matrix is nor.singular at any time t, which merely rephrases the linear independence of p.(t),....,p„(t). In terms of this fundamental matrix fi(t) the solution to (2.9) is g;iven by -1 p(t) = 8(c)C '(t=to)po (2.24) If the matrix A has N eigenvalues A.,....,Xj. with corresponding eigenvectors u ,....,u , the choice of the linearly independent solutions is particularly simple. We can take (k) pk(t) - u exPakt) (2.25) However, since A is not symmetric, there is no guarantee that there can be found N linearly independent eigenvectors correspon- ding to the N eigenvalues if degeneracies occur in the eigenva- lues. The most general form of the solution for (2.9) can be writ- ten as N p.(t) - Z c.u|k)QJk)(t)«p(A.t) (2.26) 1 k=, K l i K Here the Q,00. ,(t) are polynomials in t and the c. are arbitrary 1 (fc) constants. Regarding the Q; (t) the following cases are possible: 1 IV) 1. No degeneracy: all A, are different. We can take Q. (t) = 1. 2. Degeneracy: among the A, there is a group of m equal eigen- values, (a) If for the m degenerate eigenvalues an equal number of linearly independent eigenvectors can be found we again can (kï take Q. ' (t) =1 for this group. (b) If for the tn degenerate eigenvalues the number of linearly (k) independent eigenvectors is less than m, the 0. (t) are polyno- 11 miais of degree less than m for this group. Regarding the eigenvalues A, of the matrix A it can be shown that (2.27) For more properties, details and proofs the reader is refer- red to the paper by Schnakenberg (Sc76), where also part of the material, covered in this section can be found. 2.3. APPLICATION TO (n,y)- AND (H.I..xny)-REACTIONS Consider a number of nuclear levels, labelled 1,2,....,N with decreasing excitation energy E.»E-*....iE^. (See fig.2.2). KVI 1281 - P, 2 P2io> 3 p.») p.(0) PN(0) Fig. 2.2: Enumeration of the excited states of a nucleus as used in the master equation method, decribed in the text. T.. denotes the transition probability for going from level i to level J. Level N is the ground state. The deexcitation process of these levels is governed by a homogeneous master equation (2.9), where now T..*0 for i>j, i.e. the WxN-matrix A now reads 12 -Tl 0 0 12 (2.28) T3H' ,-rN where (2.29.a) r1. z T.. (2.29.b) This situation occurs in (n,v)- and (H.I.,xn>)-reactions. For the actual calculation of the gamma deexcitation spectra we have chosen to divide the set of nuclear levels into two distinct categories. (See Chapters III and IV). The first category contains the highly excited states, which are experimentally unknown. This category of levels is treated statistically, by using a level den- sity formula, and the gamma spectrum arising from transitions with- in this category and transitions from this category to the one des- cribed hereafter, is calculated using a cascade-technique, as will be explained in Chapter III. The second category of nuclear levels contains all the states that we take into account explicitly in our calculations. For (n,y)-reactions this category consists of all (collective) states below a cut-off energy and in the case of (H.I.,xnv)-reactions we take into account explicitly a few ro- tational bands near the yrast line. The quantum numbers of these states and the transition probabilities between them are taken from a model calculation, although, in principle,they can be ta- ken from experiment. It is to the second category of nu- 13 clear levels that we apply a master equation technique. The ge- neral idea is, that the population, which was originally distri- buted over the states in the first category, is redistributed, after a number of cascade steps, over the levels belonging to the second category. These calculated statistical populations of the explicitly treated levels now serve as the initial conditions for the solution of the master equation (2.9), describing the deexci- tation process within the second category of states. For the calculation of the total gamma spectrum only the time-integrals over the various populations are needed. They are denoted by w. w. - ƒ p.(t)dt (i-1 ,N-1) (2.30) 1 0 x Assuming that r. * 0 (i-1, ,N-1) (2.3!) the solution for the w. is easily found to be given by the recur- rence relations (2.32.a) w. =~{p.(0)+ I T..w.] (i-2 ,N~!) (2.32.b) 1 ri x j By multiplying the w. by the various transition probabilities T.. (j>i) we obtain the total time-integrated gamma spectrum. The special form of the matrix A, given by (2.28), enabled us to write down the explicit solution for the quantities w.. For the more general cases, as discussed in section 2.2, one has to integrate the solution of the master equation, as given by (2.24) or (2.26), in order to obtain the w. and thereby the spectra. For an analysis of feeding times and lifetimes, as they are measured e.g. by Bochev et al. (Bo76), it is also useful to have 14 a time-dependent solution to eq, (2.9). If none of the eigenva- lues of A, which are exactly the diagonal elements ~r., in (2.28) is degenerate, the time-dependent solution to (2.9) is given by the recursion relations p.(t) - E a..p.(t) + e. (2.33) j where the expansion coefficients a.. (j a. [ E 1 1 (2.34) 1 •V «ik**" *! and the coefficients B. are determined by the initial conditions - E a..p.(0) (2.35) J J For the time-integrals over the populations from time t=0 to t=x, we find the quantities W.(T), defined by T W.(T) = ƒ p.(t)dt (i-1,2 ,N-1) (2.36) which are explicitly given by the recursion relations WJ(T) l-exp(-r]T)] (2.37.a) ei W.(T) = E a..w.(T) +7^11 - exp(-r.T)] (i=2,..,N-l) (2.37.b) j By multiplying the W.(T) by the various T.. (j>i) we get the gam- maspectrum, as it has accumulated during a time T since the be- ginning of the deexcitation process. As an illustration, consider the time-dependent deexcitation process of a set of 200 (nuclear) levels. Let the excitation ener- gies of these levels be given by 0 3 Ei - (201 - i) * x 1.2 (MeV) (i=l 200) (2.38) 15 and let the transition probability between level i and level j (i 5 1 •I\. = i - E.) (sec" ) (2.39) where c is some arbitrarily chosen constant. In (2.38) the level density increases with increasing excitation energy, whereas (2.39) corresponds to the assumption that the transitions are of E2 single particle type. For the initial populations of the levels we take a gaussian distribution 2 2 p.(0) « exp[-(Ei-E0) /(AE) ] (2.40.a) EQ - 5.5 Mev (2.40.b) AE = 1.2 MeV (2.40.c) The gamma spectra, as they have accumulated during the first T seconds of the deexcitation process, are shown in fig. 2.3 .for 10 different values of T. There is a factor of 10 difference be- tween the accumulation times associated with two consecutive gam- ma spectra. The time-scale involved is defined by T . , which is the half-life time of the shortest living level, which is in this case the level with highest excitation energy. As can be seen from fig.2.3, the gamma spectrum starts building up its quasi-continuum part first, arising from the many possible transitions between the high-lying levels, whereas the discrete lines on the low-energy side of the spectra, arising from transitions between the well separated low-lying levels, become visible only after s3xlO T . . The total deexcitation process takes about 10 xx . . min -19 -18 In practicar l situations T nu.n is in the order of 10 -10 sec. This means that a total deexcitation process "ould take some -12 -11 10 -10 sec, which is (still) beyond the reach of present day measuring techniques. However, if one or more isomeric states in the final nucleus are present, they will delay the deexcitation process appreciably. In such a case t'e present model may be useful for the analysis of feeding times and lifetimes of these states. 16 KVI 1339 IO 0 1 2 3 4 0 3 4 E(MeV) Fig. 2.3. (to be continued) 17 I I I I Z T=3xl0 Tmin T=3xlO IOW 10 3 T=3xlO rmmiin T=3xl0 rmin 10 10 Fig.2.3. (to be continued) 18 KVI1341 10 I 4 0 E(MeV) Fig. 2.3.: The gcama spectra as they have accumulated during the first T seconds, as indicated in the figurest of the deexcitation process defined by equations (2.38)t(2.39) and (2.40). There is a factor of 10 difference in T between two consecutive figures. The last figure represents the total time-integrated gamma spectrum. T . denotes the half-life time of the shortest living levelt which is in this case the highest excited one. 19 CHAPTER III CALCULATION OF NEUTRON CAPTURE GAMMA-RAY SPECTRA 3.1. SURVEY OF PREVIOUS WORK The first schematic estimate of the spectral distribution of the radiation, emitted immediately after the neutron capture, the "primary" spectrum, was given by Blatt and Weisskopf (B152). The spectral distributions of the second gamma-rays, the "secondary" spectrum, the third gamma-rays, the "tertiary" spectrum, and so on, were estimated by Kinsey (Ki57). These partial spectra were calculated using a simple level density formula, rising with ex- citation energy as exp(2/aË), assuming only dipole-radiation with gamma-ray transition probability varying as the cube of the tran- sition energy. Originally the main theoretical interest in (n,y)-reactions was focussed on the interpretation of isomeric cross-section ra- tios, i.e. the relative probability of forming each member of a pair of nuclear states in neutron capture reactions. In 1960 Hui- zenga and Vandenbosch (Hu60, Va60) proposed a simple statistical model, using the following assumptions: 1. The nuclear levels down to the ground state can be described by a level density formula; 2. The transition probabilities between the levels are proportio- nal only to the spin-dependent part of the level density; 3. All electromagnetic cascades consist of a fixed number of transitions with muitipolarity one, the last transition feeding that nuclear state, which requires the smaller spin change; no parity changes are taken into account. Assuming that the spin-dependent part of the level density is the one predicted by Bethe (Be37), i.e. (2J+l)expl-(J+è)2/2o2J (3.1) 20 Huizenga and Vandenbosch found that their calculations were con- sistent with a spin cutoff factor o < 5. Moreover their predictions of isomeric cross-section ratios were found to be useful as a guide for assigning spins to the compound states formed in thermal or re- sonant energy neutron capture. Troubetzkoy (Tr6l) developed in 1961 a statistical theory of gamma-ray spectra following nuclear reactions. The assumptions he used differ from those of Huizenga and Vandenbosch in two respects: the nuclear levels are now described statistically only above an energy limit E~; below this cutoff energy the levels are taken from experimental data and the transition probabilities for transitions within the level continuum are taken to be proportional only to the energy-dependent part of the level density. An interpretation of isomeric cross-section ratios in terms of Fermigas and supercon- ductor models was given by Vonach et al. (Vo64, Bi64). In 1966 Ponitz (Po66) combined and extended the models of Huizenga and Vandenbosch (Hu60, Va60) and of Troubetzkoy (Tr61) to his socalled cascade model. A similar model was proposed by Von Egidy (Eg69), who considered also other multipolarities than El in the statistical cascade, using interpolated hindrance factors, taken from the work of Lobner (LÖ68). These and other models were extensively used in the determination of the spin cutoff factor o (Sc65), the interpretation of isomeric cross-section ratios (Sp67, Ma67, Sp68), the determination of neutron resonance spins (We70) and for the spin assignment of bound levels, populated in (n,y)-reactions (Br74, Co74, Br75). Population systematics of ro- tational bands in deformed nuclei was experimentally observed by Casten et al. (Ca75) and by MacPhail et al. (Ma75). A comparison of the results of calculations, using the above mentioned models, with experimental data indicates at least a qualitative agreement. In particular, isomeric cross-section ra- tios can be predicted quite well. There are, however, discrepancies with experimental results. For instance, the average multiplici- ties calculated by Pönitz (P066) seem to be systematically too low. 21 For a more quantitative understanding of the deexcitation process following neutron capture a more refined theory is required. The new features appearing in the present calculation are the following: 1. We classify the highly excited states not only according to their energies, spins and parities, but also to the numbers of unpaired neutrons and unpaired protons, thus taking into ac- count the internal structure of this class of states. 2. The low-lying states, that we take into account explicitly, can either be taken from experiment or from a model calculation. The transitions between these levels are calculated using a master equation technique. 3. Instead of taking a E^-dependence of the Ê1-transition pro- babilities for transitions from the continuum of levels to the low-lying states, which does not agree with average resonance capture data, we take into account the influence of the giant dipole resonance (GDR), as suggested by Brink (Br55) and Axel (Ax62). 4. We calculate multiplicity distributions of all gamma-rays and of gamma-rays in coincidence with certain discrete transitions. An outline of the present model will be given in section 3.2 and the details will be treated in sections 3.3, 3.4 and 3.5. Fart of the material, presented here, has been published before. (Ko78a, Ko78b). The results of all calculations, using the present model, re- 149 fer to the case of thermal neutron capture in Sm. The experimen- tal data were taken from the work of Smither (Sm66) and from Nu- clear Data Sheets (Ba76). 3.2. OUTLINE OF THE MODEL We divide the set of nuclear levels below the neutron capture state, which is located at an excitation energy equal to the se- paration energy for a neutron in the composite nucleus, into two 22 K VI1280 TT capture stole up p s s , 'Statistical cascade tewei contmuurr. E, 1 T i Q low-lying states ground stole Fig. 3.1.: The decay of the neutvon aapture state at an excitation energy equal vo the separation energy for a neutron (S^) in the composite nucleus. Indicated are primary (p)3 secondary (s)t tertiary It) and quaternary (q) transitions and some discrete transitions between low-lying levels below the cutoff energy E. categories (see fig. 3.1). The first category contains the low- lying (collective) states, which are taken into account explicitly in our calculations. The second category of excited states con- tains all the remaining levels below the capture state. Both ca- tegories may overlap each other in excitation energy. In applying our procedure to the reaction llt9Sm(th.n,Y)150Sin we took into account explicitly all the collective states below a cutoff energy E (E »2 MeV), calculated with the Interacting Boson Model of Ariraa and Iachello (Ar75, Ar76, Ar78a, Ar78b; see also Appendix A). The level continuum is treated in a statistical way, by using a level density formula. For the energy-dependent part of the level density we used the expression of Ericson (Er58), where- as for the spin-dependence of the level density the parametriza- tion, proposed by Bethe (Be37) and Bloch (B154) and extensively discussed e.g. by Gilbert and Cameron (Gi65a, Gi65b), is used, eq. (3.1). The actual calculation of the total deexcitation spectrum following the neutron capture is performed in two steps: 23 1. In the first part of our calculation we compute the statisti- cal decay of the capture state using a cascade technique. The first cascade step gives rise to the primary gamma spectrum, the second cascade step to the secondary spectrum, and sc on. In each cascade step the total remnant population, present in the continuum of levels, is partly redistributed over the level continuum and partly relocated in the low-lying levels. After a number of cascade steps convergence will be reached in the sense that the total population, that was originally concentrated in the capture state, has completely been redistributed over the low-lying states. These populations will be called the statis- tical populations of the low-lying levels. Although, in prin- ciple, we could have calculated the statistical spectrum with a master equation technique as well, we chose not to do so, main- ly because the cascade technique gives us immediately the multi- plicity distribution of the statistical feeding of the low-lying levels and their statistical populations. The master equation technique does not allow a simple extraction of these physi- cal quantities, which can be compared to experimental data, but is computationally faster. 2. In the second part of our procedure we calculate explicitly the transitions between the low-lying states, showing up as dis- crete lines in the gamma-ray spectrum riding on a continuous background of statistical gamma-rays. For this purpose we use a master equation technique, as described in chapter II. The word "statistical" played a keyrole in the previous para- graph. It should be pointed out here what exactly is meant by it in the present context. For some time it was believed that the radiative capture process is of purely statistical nature and that the capture state could be well described by the Bohr com- pound nucleus hypothesis. However, the discovery of size resonan- ces in average neutron cross-sections led Feshbach et al. (FeS4) to the interpretation that the captured neutron retained its identity for some time. Later work (B163) showed the existence also of more 24 complicated doorway states, forming a hierarchy of increasingly complicated excitations, eventually leading to the compound nucleus. For a more recent review of the present status of the theory of the capture mechanism the reader is referred to e.g. (Ch77, Le77, Mo76). The non-statistical nature of the composite nucleus forma- tion in neutron capture reactions has important consequences for the decay of the capture state. In the case of thermal or reso- nant neutron capture, the capture state is dominated by one speci- fic resonance state. The compound nucleus hypothesis would imply a smooth energy-dependence of the intensities of primary transi- tions. On the contrary it appears that these intensities are in some cases strongly correlated with the spectroscopie strengths in a (d,p) reaction, leading to the same final states, demon- strating a direct capture effect (Ko74, Sp74). For our purposes, however, one can safely assume that the decay of the capture state is statistical, because, in the total gamma spectrum, the non- statistical components will be completely washed out by the over- whelmingly large number of intermediate states, through which a gainna cascade can proceed. On the other hand, the calculated in- tensities of the high-energetic primary transitions directly to the individual low-lying levels can be compared to the experiment- al data only on an average. For the energy-dependence of the transition probabilities for transitions from the continuum to the low-lying states we use the Brink-Axel hypothesis. Details will be given in section 3.3. For transitions within the level continuum the single-particle estimate for the transition probabilities is used. As mentioned before, throughout the statistical cascade we keep track of the multiplicity of the statistical feeding of the low-lying levels. Combined with the non-statistical feeding and the multiplicity distributions of gamma-rays deexciting the levels, we are able to calculate the multiplicity distributions of gamma-rays in coincidence with some discrete transitions. For (n,y)-reactions a comparison of measured and calculated multi- 25 plicity distributions can yield important information on the multi- polarity of the statistical gamma-radiation, as will be discussed in section 3.5. 3.3. THE STATISTICAL CASCADE For the actual calculation of the gamma-cascade deexciting the capture state we need the following ingredients: 1. a level density formula describing the highly excited states; 2. initial population, representing the capture state; 3. transition probabilities within the level continuum; 4. transition probabilities for transitions from the continuum to the low-lying states; 5. spins, parities and excitation energies of the low-lying levels. These ingredients will be discussed in detail successively now, except the quantum numbers of the low-lying states. They will be discussed in Appendix A. 3.3.1. Level Density Formula. For the energy-dependent part of the level density formula, describing the level continuum, we used the Ericson expression (Er58), classifying the states according to their excitation energies (E), the number of unpaired neutrons (n) and unpaired protons (z). The partial level densities are given by p(E,n,z) (3.2) Here A denotes the pairing energy and u is related to n and z by |(n+z) (even-even nuclei) i(n+2-l) (odd mass nuclei) (3.3) J(n+z-2) (odd-odd nuclei) 26 p is given by the Ericson formula (Er58) Pj(U,n,z) = gn-l/2 g«-l/ (3.4) lfr n!z!2[ {f gU1+(^L With (3.5) Formula (3.4) is derived by considering a system of non-inter- acting neutrons and protons, allowed to occupy equidistant single- particle levels. The single-particle level densities are denoted by g, and g, for neutrons and protons respectively. Each single- particle level can accommodate at most two nucleons of one kind (spin up and spin down). Let the single-particle energies for neutrons and protons be denoted by a and b respectively and let the corresponding occupation numbers be n and z (n , z = 0, 1, s s s s 2). Expression (3.4) represents the number of distinct distri- butions of neutrons and protons over the single-particle levels per energy interval at an excitation energy U, subject to the following restrictions: U = £ n + E z b (3.6) s s s s s N - £ n (3.7.a) s s Z - £ z (3.7.b) s s n * £ «n (3.8.a) z • £ (3.8.b) ss» where N and Z are the total number of neutrons and protons, v and g are abbreviations for n+z (3.9) 27 gl+gg g = (3.10) The interaction between the nucleons is taken into account by an effective pairing energy, A, required to break up a pair of iden- tical nucleons occupying the same single-particle level. The total level density is given by p(E) = Z p(E,n,z) (3.II) It should be understood that in (3.11) all the magnetic sub- states belonging to some particular value of J are counted as levels. The neutron and proton single-particle level densities we used for Sm are the following = 4.0 MeV-1 g = 8, = g2 (3.12) The pairing energy, A, was fitted in such a way that (3.11) re- produces the total level density of Gilbert et al. (Gi65a,Gi65b) at the excitation energy of the neutron capture state. We found A » 1.0 MeV (3.13) For the spin-dependence of the level density ve took the usual form f(J) (3.14) 2o3/2ir The spin cutoff parameter, o, is tasten from the work of Gilbert et al. (Gi65a,Gi65b). For >50Sm we have O-5.1 (3.15) From (3.14) it can be seen that 28 Z (2J+l)f(J) - I (3.16) J We assume equal numbers of positive and negative parity sta- tes in the level continuum. That this is a reasonable assumption can be seen from the following arguments. Consider a state made up of k single-particle excitations. Let p+ be the probability for one fermion to occupy a positive parity state and p_ = 1-p the probability for one fermion to occupy a negative parity state. The probability of having j out of k fennions in negative parity states and the others in possitive parity states is , .k-i j kl (3.17) so that the total probability P_ of having a negative parity sta- te for the nucleus is given by k' .(in (3.18) j odd k 3)' and the total probability P of having a positive parity state j even J The last equality follows from the binomial theorem. We can eva- k k luate (3.IP) and (3.19) as follows. First evaluate P+ -P_ (3.20) k k Now we can solve P_ and P from the last equality in (3.19) and (3,20). We find (3.21.a) P K = + 2 (3.21.b) 29 From these expressions we see that for the larger k-values, i.e. for the higher excitation energies, the probability for a state to have either parity is almost equal to £, unless there is strong reason to believe that p is very close to either 0 or 1. The total density of states at excitation energy E, having spin J and parity ir (+1 or -1) is now given by P(E,J,TT) Z p(E,n,z) = if(J)p(E) (3.22) n.z Here the m-degeneracy is taken out, and we see, using (3.16) that Z (2J+1)P(E,J,TT) = p(E) (3.23) J,TT vbich explains the remark, following (3.11). In fig. 3.2 we show plots of the various partial level densi- ties p(E,n,z) as functions of excitation energy for the nucleus Sm. The parameters of (3.12) and (3.13) are used. As can be seen the curves are steepest for the highest values of the total number of unpaired particles, v. 3.3.2. Transitions within the Level Continuum. For the explicit calculation of the gamma cascade we divide the continuum of levels into a number of bins, each bin being characterized by the five quantum numbers E,J,ir,n and z. We assume only El-transitions in the statistical cascade. For transitions wi-hin the continuum, from one bin to another, the following selec- tion rules should be obeyed: - 0,1 (3.24.3) = -1 (3.24.b) |An|,|Az| * 0,2. Not I AnI | Az | (3.24.C) This last selection rule, which is due to the one-body character 30 ?•" 10 Fig. 3.2.: Partial level densities o(E,n1z.- - aiE^z^n; saloulated h the Evioson formulas (2.2) - f?..5), •'5.9)a (1.10). The para- l ! used are g1 - gr = 4.0 MeV and a - J.'. MeV. The various '«,3/ combinations are indiaated on the right. The full aurve 3hows the sum of all partial level densities^ ec. 's.ll). of the electromagnetic transition operator, was ignored in pre- vious models. It reflects the internal structure of the level con- tinuum. Froir fig.3.2 it can be seen that its effect will be a sup- pression of the high-energetic transitions, which would energeti- cally be favoured, but predominantly require a violation of the third selection rule. The introduction of the quantum numbers n and z immediately raises the question of how to choose the initial population of the level continuum, so as to represent the capture state. In 'Sn: mi? state, formed by thermal neutron capture, is dominated 3! by a nearby 4 resonaace and its excitation energy is 7.9861 MeV, being the separation energy for a neutron irom Sm, i.e. E.. = [M( Sm) + M(n) - M(15OSm)]c2 = 7.9861 MeV (3.25) This capture state, with its excitation energy sharply determined and with definite spin and parity, will be built up of several components, containing different numbers of unpaired neutrons and protons. Without any a priori detailed knowledge of the wave function of the capture state it is reasonable to distribute the initial population, representing the capture state, over the bins with E = 7.9861 MeV and Jv = 4~ proportional to the partial level densities p(E = 7.9861,n,z) as appearinappearing in the Ericson expres- X sions (3.2)-(3.5),(3.9) and (3.10), i.e. p(E =7.9861,J1T=4~,n,z) = X (3.26) = P p(E =7.9861,J1I=4",n,z) / 1 p(E =7.9861 ,Jïï=4~,n',z' x . • x n ,z and all other initial populations are equal to 0. P represents here the total initial population. For the energy-dependence of the EI-transitions within the 3 continuum we take the usual E -form, i.e. for the partial decay probability of a level, within a bin characterized by E,J,tr,n,z to all levels within the continuüm we take: J rc(E,J,it,n,z) B x (E-Ef) N(Ef,Jf,irf,nf,zf)(3.27) where the summation runs over all final bins with Ef N(Ef,Jf,TT-,n,,zf) denotes the number of levels within th*> bin characterized by the five quantum numbers, appearing as the argu- ments of N. The subscript on F _ stands for continuum. B is a con- stant, containing the. average reduced matrix element for EÏ -fran - 32 sitions. Since we are not interested in the absolute time-depen- dence of the statistical cascade, the actual value of B is imma- terial. For the calculation of the total radiative decay width of a level within a bin characterized by (E,J,Tr,n,z) we must add to (3.27) the partial decay width for decay to the low-lying levels. This will be done in the. next subsection. 3.3.3. Transitions from the Level Continuum to the Lov7-lying States. We have taken special care in treating the energy-dependence of transitions from the level continuum to the low-lying states correctly. Average resonance neutron capture data indicate that the energy-dependence of primary El-transitions is not E , but rather E^ for the high-energy gaiamas (see e.g. Bo6?,Bu70). We shall illustrate this point for the case of average resonance cap- 149 149 ture on Sm. Observing that the ground state spin of 'Sm is 7/2 and assuming only s-wave ntutron capture, the resonance spin can either be 3 or 4 . The primary El-transitions fall into two categories (See fig.3.3). Ei 149 Fig. 3.3.: In average neutron resonance capture on Sm the spin of the resonance state can either be S or 4 . The primary El- transitions fall into two categories: those feeding 3 or 4 final states and those feeding 2 or 5 final states. The intensity ra- tio of those categories is roughly equal to 2:1, being the ratio of the number of ways the final states in the two categories can be reached. 33 KVI1283 I i ... I i ...! i ... I .... I 6 I 5 3 4 -uï" '« 2 UJ 5.5 6 6.5 7.5 Ey(MeV) Fig. 3. 4.a. (Upper part): Intensities of primary El-transitions 3 149 divided by E for average resonance capture on Sm. The data were taken from (Ba?6). Transitions leading to final states with (tentative) spins 2 or 5 are denoted by triangles and with (ten- tative) spins 3 or 4 by circles. The transition between paren- theses involves probably more than one final state. The average values of the two categories of reduced intensities are indicated by the two dashed lines. 3.4.b. (Lower part): Intensities of primary El-transitions •z o 149 divided by E /(ER - E ) for average resonance capture on Sm. 34 One category contains the primaries leading to final states having spin 3 or 4 , which can be reached by El-decay of both 3 and 4 resonances. The second category of primary transitions feed either a 2 state (from the decay of a 3 resonance) or a 5 state (from the decay of a 4 resonance). The intensities of the transitions belonging to the first category are roughly a factor of 2 higher than those of the second category. If the energy-dependence of 3 these primaries were E , then, by dividing the intensities by this factor, we would expect to find two classes of practically constant reduced intensities. This is not the case as will be clear from fig. 3.4.a. There is a systematic trend of increasing reduced in- tensities with larger gamma-ray energies. A phenomenological explanation for this deviation from the expected E -dependence was civen by Brink (Br55) and Axel (Ax62), who took into account the influence of the GDR. We shall illustrate 149 their explanation in the case of neutron capture in Sm. Two ba- sic assumptions are made: 1. The El-decay of a state i with J = 1 directly to the ground state proceeds via the component in the wave function of state i, that represents the GDR strength present in this state, i.e. (see fig.3.5.a) I >. = c.(I )[GDR> + other components (3.28) 2. The £1-decay of a state i, say with J = 4 , to a low-lying state f with e.g. J - 3 proceeds via the component in the wave function of state i, describing a GDR built on top of state f, i.e. (see fig. 3.5.b) !*"*£ = ci(4)|f3*® GDR]4~> + other components (3.29) This last assumption is usually referred to as the Brink-Axel hy- pothesis. Now the absorption cross-section of a photon of energy 35 KVI1285 -'obs ER*E, Fig. 3.5.a. (Upper part): The El-decay of a 1~ state directly to the ground state proceeds via the component in its wave function describing the admixture of the GDR into it. The curve represents the absorption spectrum of photons incident on the target in its ground state. 3.5.b. (Lower part): The Brink-Axel hypothesis: the El-tran- sition from state i with Jv = 4~ to a state f with Jv = 3* proceeds via the component in the wave function of state it describing a GDR built on top of state f. The curve represents the fictitious ab- sorption spectrum of photons incident on the target ir. the excited state, described by the wave function \3 >f. It is the same curve as in fig.3.5.at but shifted over E^. towards higher energies. 36 E * he/ft by the nuclear ground state (J* • 0 ) to a single isola- ted level with j" « 1 and excitation energy EQ is given by (3.30) [(2/D(E-E0)J-+l where T is the full decay width of the excited state and rfl is its partial width for deexcitation to the ground state. The subscript n denotes the natural line shape energy-dependence of (3.30). The experimentally measured photon absorption cross-section is an e- nergy average of (3.30), i.e. Oa(E) (3.31) Here D denotes*the average 1 level spacing at excitation energy E. Axel (Ax62) parametrized the shape of the GDR, observed in the photon absorption process, by a Lorentz shape (1.3A/100r )E2r2 a (E) (b) (3.32) a * 2 2 2 * 2 2 a (Eg-E2)S EV where F , the full width at half maximum of the GDR, E , the cen- g KD troid energy of the GDR and E are given in MeV. By equating (3.31) and (3.32) Axel obtains, at an excitation energy E * 7 MeV -(E = 7 MeV) cc E" (3.33) D while for E s 3 MeV 3 MeV) « E (3.34) The main disadvantage of this phenomenological explanation is, that the shape of the GDR, as parametrized in (3.32), has to be extrapolated to very low excitation energies, compared to the cen- troid energy and the width of the GDR. It is questionable that (3.32), which decribes the overall shape of the GDR reasonably 37 well, is also suited for the description of its tail. 3 In a more microscopic picture the departure from the £ -depen- dence can be understood by calculating in a statistical way the spreading of the GDR over the many underlying I'states (see e.g. 3o67a). For the expansion coefficients c.(l ) in (3.28) one obtains (3.35) where v represents the average matrix element of the Hamiltonian connecting the GDR and the states i. In the same way the expansion coefficients c.(4 ) appearing in (3.29) can be obtained by shif- ting the entire picture over an amount Ec towards higher excita- tion energies, i.e. (3.36) c. ) - WEi Here E, denotes the energy of the final state |3T>..By noting that the energy of the gamma, associated with the transition from state i to state f, is given by E - E.-E, (3.37) Y if we obtain c.(4 (3.38) For the partial El-decay width of a 4~ level to a 3 level we find then, by invoking the Brink-Axel hypothesis «OX — I _ /ƒ \ 1 ^ T* "^ f3.39) 38 If we would approximate (3.39) by » const. E (3.40) the power p is energy-dependent and is given by E (3.41) For E =7 MeV and E^ - 14.6 MeV (Ca74) we find p(E ) = 4.8. It is interesting to note that in this simple model we find (3.42) contrary to Axel's result: p(E =0) 4 (See 3.34). In order to see how well (3.39) describes the energy-depen- dence of the primary El-transitions in the case of average reso- 149 nance capture on Sm we show in fig.3.4.b a plot of the intensi- 3 2 ties of these transitions divided by a factor E /(E -Z ) with "R E_ * 79A.-1/ "J3 MeV. It can be seen from this figure that the redu- ced intensities clearly separate into two distinct, practically constant classes. For the partial Ei-decay width of a state, characterized by the quantum numbers (E,J,ir,n,z) to all low-lying discrete states, we are therefore led to the following expression. „ 3 rd(E,J,iT,n,z) (3.43) f The total decay width of a level with quantum numbers (E,J,ir,n,z) will now be given by the sum of expressions (3.27) and (3.43), i.e. r(E,J,ir,n,z) - rd(E,J,ir,n,z) (3.44) 3.3.4. Results. In this subsection the results of a calculation of the statis- tical gamma cascade will be presented for the case of 149Sm(th.n,Y)l5°Sm. In fig. 3.6 a comparison is shown between the experimental and theoretical statistical populations of the levels below 2 MeV in I en Sm. The experimental points (open symbols) were deduced from the data (Sm66,Ba76) in the following way. For a particular level all intensities of the transitions deexciting this level were added. From this result all intensities of transitions between other sta- tes below 2 MeV feeding the level under consideration were subtrac- ted. The result is the net population of this level, due to the sta- tistical gamma cascade. The lines in this figure connect the theore- tical points (filled symbols) belonging to states with the same spin and parity. The fact that there is no one-one correspondence be- tween experimental and theoretical points is due to the fact that for the excited states we used the results of a theoretical calcu- lation using the Interacting Boson Model of Arima and Iachello (Ar75,Ar76,Ar78a,b,see appendix A). The agreement between theory and experiment is quite good, both for positive and negative parity states, with the exception of the 0 -states, the populations of which are calculated too low. This is due to the fact that E2-tran- sitions in the statistical cascade are completely ignored. A small £2-admixture would allow a feeding of the Q -states in the third cascade step already. Taking into account only El-transitions im- plies that this feeding can take place only after 5 cascade steps (because of angular momentum and parity considerations), by which time the cascade process has mostly converged. 40 10 - 1.5 ExlMeV) Fig. 5.6.: Logarithmic plot of experimentally determined (open sijmlols) and theoretically calculated (filled symbols) populations of zhe even parity (left) and odd parity ivigkt) states below 2 MeV versus excitation energy for the reaction ' Sm(th.n,y)* 3m. The meaning of the various symbols is indicated in the figure. The lines connect the calculated points for the 0 S2 and 4 (left) and the 1 and S (right) levels. The slopes of the lines connecting the 0 ,2 ,4 ,1 and 3 states in fig. 3.6 are sensitive to the actual value of n in (3.43). It is easy to see that the smaller the value of n is, the later the population rains into the low-lying levels, and hence the steeper the slopes are. A fit to the experimental slopes yields for n in this case 25(±3) MeV' (3.45) 41 It is interesting to see a decomposition of the calculated statistical gamma-ray spectrum into its partial spectra. Fig. 3.7 shows the calculated primary spectrum, the accumulated gamma spec- tra after two and three cascade transitions and the total statisti- cal gamma-ray spectrum. KVI 1278 l4"Sm(th.n,y)ts0Sm statistical 10 :sjv ^~ P »'~^jv — P*S r "^s^s "Si' total 10° w r j lö-'lf Jf 4 t : : 01 ,12 3 4 5 6 \J7 Fig. 3.7.: Calculated decomposition of the statistical part of the 149 ISO gamma-ray spectrum in the reaction Sm(th.nty) Sm. Shown are the primary spectrum (p)t the accumulated spectrum after two cas- cade transitions (p+s)t the accumulated spectrum after three cas- cade transitions (p+s+t) and the total statistical garma-vay spec- trum (full curve). For n the value 25 Mev was used. The peak energy of the primary El-spectrum is calculated at E • 1.65 HeV. This is much lower than the corresponding value estimated by Blatt and Weisskopf (B152) and Kinsey (Ki57). They obtain for this peak energy a value which is about a factor of two higher. This difference can be explained partly by the fact that they used a different energy-dependence for the level densi- ty and partly by our additional internal structure of the level continuum (as expressed by the quantum numbers n and z), which 42 suppresses the high-energy transitions, as was explained in sub- section 3.3.2. The discrete peaks in the high-energy part of the statisti- cal gamma-ray spectrum correspond to primary El-transitions di- rectly from the capture state to the low-lying levels. Because of the non-compound nature of the capture state, the intensities of these individual transitions can be compared to experimental data only on an average, as was explained in section 3.2. This, of course, does not imply that the bulk of the calculated statisti- cal spectrum cannot be compared to experimental results, but, be- cause of the lack of sufficiently accurate data, we have not done so. 3.4. THE DISCRETE GAMMA LINES Once the statistical gamma cascade has converged and the to- tal population, that originally represented the neutron capture state, has completely been redistributed over the low-lying le- vels, these statistical populations can be used for the calcula- tion of the discrete transitions between the low-lying states. These transitions appear in the total gamma spectrum as discrete lines, riding on a continuous background. To this end we use the master equation technique, explained in section 2.3. The various transition probabilities between the levels are calculated using the Interacting Boson Model. We took into account only El- and E2-transitions. For more details the reader is referred to (Ar75, Ar76,Ar78a and b) and to Appendix A. In fig.3.8 the calculated time-integrated gamma-ray spectrum, arising from transitions between states below 2 MeV in Sm, for 149 the case of thermal neutron capture on Sm, is compared to ex- perimental results (Sm66,Ba76). The agreement between experiment and theory is good. Especially the rise of intensity in the first 500 keV gamma-ray energy over more tnan three decades is reproduced very well. Moreover, sines ve used tor cur calculations Fig. 3.8.: Comparison of the measured (Ba?6) (full bars) and calculated (dashed bars) gamma spectrum, arising from the de- excitation of the levels below 2 MeV in Sm in the reaction M9 1S0 Sm(th.nty) Sm. 0 .5 I 1.5 2 Ey(MeV) theoretical values for the excitation energies of the low-lying levels and therefore also for the gamma-ray energies of the va- rious transitions, it cannot be expected that the calculated curve follows the experimental one in every detail. The agreement be- tween experiment and theory is even more striking if we compare the calculated and measured intensities of individual transitions (Table 3.1). In particular the high intensities are reproduced very well, except for the 3. to 2 transition, which is due to the fact that the calculated value for the statistical population of the 3. - state is about a factor of two higher than the corres- ponding experimental value (see fig. 3.6). In fig.3.9 the calculated total (i.e. statistical plus dis- crete) gamma-ray spectrum is shovn for the reaction Sm(th.n,y) Sm. Because no sufficiently accurate data are a- vailable, we did not compare this spectrum to experimental results. 3.5. MULTIPLICITIES AND MULTIPLICITY DISTRIBUTIONS Because of the growing interest in multiplicities and roulti- 44 Table 3.1. Transition E (exp.) Ey(th.) A(exp.) I(exp.) Kth.) (keV) (keV) a) ot 333.95 355.9 E2 64.2 64.2 2* 439.40 423.4 E2 35.8 36.9 et -»• 4* 505.41 489.0 E2 6.7 5,1 8Ï* 6* 558.25 .044.0 (E2) (.23) .05 77 72.33 99 8 .0008 02 •* 2Ï 406.44 402.6 E2 .43 1.02 2j* 0* 1046.6 971.2 (E2) .23 .67 2Ï* 2t 712.14 615.3 MK+E0+E2?) 3.07 3.05 2Ï* 4t *272.83 191.9 .043 .002 42 -> 2* 1115.22 1125.8 .085 4^ *1 675.82 702.4 E2+M1+E0 1.99 1.32 42 * 61 170.23 213.4 (E2) .0051 .0010 2^ 0^ 305.68 212.7 E2 .10 .009 42 -> 22 402.99 510.5 E2 .95 .84 37 377.74 408.9 .060 .082 + 57 91.8 147.9 (El) .0038 .012 4z -> 23 255.34 200.8 E2+? .021 .001 37- 2t 737.38 716.9 El 6.56 11.6 3Ï-*- *l 298.06 293.5 El .50 .79 57 -»• *l 584.26 554.5 El 6.05 6.3 57 •»• et 78.76 65.5 El .022 .009 77 •*• et 485.87 444.2 (El) .81 .39 37 - it 25.26 101.6 .003 57 •* 37 286.28 261.0 .017 .030 77 - 57 407.16 378.7 .03 a) Normalized to experiment. Table 3.1.: Comparison of the calculated and measured (Ba76) inten- sities of gamma transitions deexdting the states of the first two 150 positive-parity and the first negative-parity band in Sm in the 149 case of thermal neutron capture on 3m. Column 1 contains the transitions3 column 2 the experimental gamma energies and column 3 the theoretical IBA-results for the gamma energies. The experimen- tally determined multipolarities are given in column 4. The experi- mental and calculated intensities are contained in columns 5 and 6 respectively. 0 2 4 6 8 Fig. 3.9.: Calculated total (i.e. statistical plus discrete) gamma 149 spectrum following thermal neutron capture in Sm. plicity distributions of gamma-rays, especially in (H.I.,xny)-re- actions, we also calculated these quantities in the case of neu- tron capture reactions. At every stage in che calculation of the statistical gamma cascade we kept track of the statistical feeding of all low-lying levels in that particular cascade step. Expressed as fractions of the total statistical feeding of the low-lying levels, they define the normalized multiplicity distributions of the statistical fee- ding of these levels. Let the intensity of the feeding of level i in the M-th cascade step be denoted by I.(M), then the total sta- tistical population of this level is given by I.(M) (3.46) M and the normalized multiplicity distribution of the statistical feeding of level i is given by (3.47.a) f.(M) 1 (3.47.b) M Due to the (non-statistical) decay of this level the multiplicity distributions for the total (i.e. statistical plus non-statistical) feeding of the lower levels j (j>i) are altered in the following way: fj(M) (3.48.a) fj(M) Z f! (M) = 1 (3.48.b) to 3 In this way the multiplicity distributions of the total feeding of all low-lying levels can be calculated iteratively. The multiplicity distribution of the total feeding of the ground state is then equivalent to that of all gamma-rays follo- wing the neutron capture. Fig.3.10 shows this multiplicity distri bution, calculated for the reaction Sm(th.n,v) Sm. The ave- rage multiplicity, 0 3 4 6 8 10 12 14 Fig. 3.10.: Calculated multiplicity distribution of all gamma-rays 149 following thermal neutron capture in 3m. Draper et al. (Dr60) measured two different values for p.^(M) = I f.(k)g.(M-k) (3.51.a) I P. ,(M) - I (3.51.b) M 3 The calculated multiplicity distributions of all gamma-rays that are in coincidence with the 2. to 0., the 4. to 2., and the 6. to 4. transitions in the reaction Sm(th.n,v) Sm are shown in fig.3.11. The interesting feature that we observe in this figu- re is the odd-even staggering of the multiplicity distributions, which is especially clear for the 6. to 4. transition. This can easily be explained if we observe that the capture state is do- minated by a nearby 4 resonance. Assuming only El-transitions in the statistical cascade, the 6. state can be reached only in an odd number of cascade steps, because of parity considerations. Furthermore, the 4. state can only decay to the ground state in two steps via the 2. state. Thus we expect minima that are rigo- rously zero for M = 4,6,8 etc. However» these minima are filled in somewhat due to the non-statistical E2-feeding of the 6. level. An experimental determination of this kind of multiplicity dis- tributions, with a multi-detector coincidence technique, is expec- ted to give valuable information on the amount of Ml and/or E2 ad- mixture into the statistical cascade. Such an admixture would fur- ther fill in these minima and furthermore it would affect the over- all shape of the total gamma spectrum. In particular an E2-admix- 49 KVIII76 4 6 8 10 12 14 M Fig. 3.11.: Calculated multiplicity distributions of all gamma- rays that are in coincidence with the 21 to 0. transition, (full curve), the 4^ to 2- transition (dashed curve) and the 6. to 4. transition (dotted curve) respectively for the reaction 149 1S0 Srn(th.nty) Sm. ture would increase the high-energy part of the spectrum, which would probably be in better agreement with experimental results. The importance of other than E!-radiation in the statistical cas- cade was already pointed out by Sperber (Sp67). 3.6. SUMMARY AND CONCLUSIONS In this chapter a new and detailed model for the calculation of gamma-ray spectra following neutron capture was presented and 149 applied to the case of thermal neutron capture on Sm. A dis- tinction was made between the high-lying states, which were trea- ted in a statistical way, and the low-lying levels, that we took into account explicitly and which were calculated using the Inter- acting Boson Model. The statistical gamma cascade, consisting of transitions 50 within the level continuum and transitions from this continuum to the low-lying states, was calculated using a level density formu- la that classifies the states according to their excitation ener- gies, spins, parities, and the number of unpaired neutrons and unpaired protons. We assumed only El-transitions in the statisti- cal cascade. The additional, as compared to previous models, inter- nal structure of the level continuum, expressed by the quantum numbers n and z, led to a new selection rule <3.24.c) for electro- magnetic transitions, which was ignored in other models. The in- fluence of this selection rule on the calculated results is in the first place a suppression of the high-energy transitions within the level continuum and secondly a calculated average multiplici- ty that is higher than previously calculated. On the other hand, the introduction of the internal structure in the level continuum raised the question of how to choose the initial population, so as to represent the capture state. In expression (3.26) we used the statistical assumption that the components in the wave function of the capture state with different n and z are excited proportional to the various corresponding partial level densities at the exci- tation energy of the capture state. However, by giving larger weights to the components with the smaller n and z, we also have the possibilitv to describe neutron capture processes with a more direct character. The resulting calculated statistical populations of the low- lying levels are in very good agreement witli experimental results, which shows the validity of the assumptions, used in the statisti- cal cascade calculation. Only a small amount of E2-admixture into the statistical cascade will improve the agreement between theory and experiment for the statistical populations of the 0 states, which are calculated too low. In the second part of our calculations we computed the indi- vidual electromagnetic transitions between the low-lying levels using a master equation technique. These levels and the electro- magnetic transition probabilities between chem were calculated 51 using the IBA-model of Arima and lachello (Ar75,Ar76,Ar78a,b), al- though in principle we could have taken them from experiment. A comparison with experimental data showed a good agreement, not only for the overall shape of the discrete gamma spectrum, but also in detail, when we compared calculated and measured intensities of in- dividual transitions (Table 3.1). This shows the validity of the Interacting Boson Model. In particular, calculated branching ratios can be compared to experimental data. In general, these branching ratios agree well with experiment, although some problems remain. Finally we discussed the calculation of multiplicities and multiplicity distributions of gamma-rays. As was shown in section 3.5, an experimental determination of the multiplicity distribu- tions of gamma-rays in coincidence with discrete transitions may provide us with important information on the amount of Ml and/or E2 admixture in the statistical gamma cascade. The proposed model is also suited for the calculation of the gamma spectra and multiplicity distributions of gamma-rays in (H.I.,xnv)-reactions. One important difference between these reac- tions and (n,y)-reactions is, that we now have to deal with a dis- tribution of initial population, not only in spin and parity, but also in excitation energy. This problem will be discussed in more detail in Chapter IV. 52 CHAPTER IV SPECTRA FOLLOWING (il.I.,xn)-REACTIONS 4.1. INTRODUCTION The earliest experimental attempts to study the gamma deexci- tation process following the neutron evaporation in (H.I.,xnv)-re- act ions were made by Mollenauer at Berkeley (Ho62), by Oganesyan and coworkers at Dubna (0g63) and by Morinaga and Gugelot at Am- sterdam (Mo63). Mollenauer used mainly a and C beams, whereas the Dubna group used 0 ions as projectiles. The experiments by Morinaga et al. were carried out with a beams. These first experi- ments essentially involved the measurement of singles gamma-ray spectra recorded with a Nal detector, which made the determination of average multiplicities, i.a. the average number of gamma-rays in the deexcitation cascade, possible. The discovery of interes- ting phenomena related to high spin states in nuclei, such as back- bending (Jo71), angular momentum-induced shape changes (Pr73) and rotation alignment (St72a,St72b), gave new impetus to research in this field. In 1974 der Mateosian and collaborators (Ma74) used two Ge(Li) counters in coincidence in order to study the continuum 19 gamma-rays in ( "C,xnY)-reactions mainly. They were able to pick out the spectrum of a particular (xn)-reaction channel by setting windows on known discrete transitions in the ground state band of the desired reaction product. Average gamma-ray multiplicities for 12 several ( C,xny)-reactions were obtained by determining the num- ber of gamma-rays in one counter in coincidence with the gate tran- sition in the other detector. Tj«5m et al. (Tj74) used a coincidence setup, consisting of a Ge(Li) and a Nal detector. Measurements of higher-order coincidences, using additional Nal counters, were car- ried out by Hagemann et al. (Ka75). In this way information about the higher moments of the gamma-ray multiplicity distribution could be obtained. This line of investigation was pursued by the 53 Copenhagen group and followed by several other groups (Ba75,Si77, An78). At Groningen experiments were carried out by Ockels, de 12 Voigt and others (Oc76,Oc77,Su77,Oc78), using mainly a and C beams from the KVI cyclotron. A multi-detector coincidence setup, consisting of one Ge(Li) and up to 16 Nal detectors, was used for the study of the continuous gamma spectra following (a,xn)- and ( C,xn)-reactions. A review of the present status of (H.I.,xny)- reactions can be found in (Di76) and (Mo76). The observed gamma-ray spectrum in a (H.I.,xnY)-reaction con- sists mainly of three parts (See fig. 4.1): Fig. 4.1: Schematic representation of the gamma-ray spectrum, emitted in a (H.I.txny)-reaction. 1. A high energy part of the spectrum, starting from about 1.5 MeV, which decreases approximately exponentially in intensity with increasing gamma energy. Measurements of gamma-ray anisotropies (Di76) and conversion electrons (Fe77) indicate that the multi- polarity of the radiation in this part is predominantly El. 2. Below - 1.5 MeV in some cases a bump appears, that extends downwards in energy. The multipclarity of the radiation in this part of the spectrum is mainly £2 and possibly Ml (Fe77). 3. On top of the bump strong discrete low-energy lines are obser- ved, arising from transitions between states belonging to the ground state band. The exponential tail of the spectrum can be interpreted as arising from statistical transitions from the entry states to a region near the yrast line, as was already shown by Grover and Gilat (Gr67a, Gr67b,Gr67c). The low-energy bump may be interpreted as being pro- duced by the gamma-rays emitted by the nucleus at high spin, on its way down to the ground state, along bands parallel to the yrast li- ne. Recently Liotta and Sorensen (Li78) analyzed the gamma-ray spec- tra in terms of a competition between collective (E2) and statisti- cal (El) dt ixcitation modes. The maximum angular momentum brought into the compound system in a heavy ion reaction can be estimated from simple considerations by 2 2 [2ii(Rp+Rt) (E-V)/h ]' max where u is the reduced mass of the system, E and V are the energy and Coulomb barrier in the center-of-mass system and R and R de- note projectile and target radii respectively. In the sharp cut-off model the partial fusion cross section, associated with a particu- lar value of t, is given by (4.2) Tfi max (4.3) max (4.4) The total cross-section is obtained by summing the partial cross- sections 55 (4.5) total (2C+1)T£ " Upon substitution of (A.I) into (4.5) we find (4.6) total This linear dependence of o . on I/E is borne out by experimen- tal results. (See e.g. Gu73,Br75a). The maximum excitation energy attained by the residual nu- cleus in a particular (xn)-reaction channel can be calculated from the binding energies, assuming the neutrons to be evaporated with zero kinetic energy. (4.7) Ema* x = E-B p -B t+ Br Here E is the projectile energy in the center-of-mass system. B , B and B denote the binding energies of the projectile, the tar- get and the residual nucleus (after the neutron evaporation) res- pectively. For the reactions that we shall consider further on in this chapter, viz.15ONd(a,4n)15OSm (E 45 MeV), 158Gd(a,4n)158Dy a (E - 40 MeV) and 160Gd(a,4n)l60Dy (E = 40 and 49 MeV), the va- lues of E* are displayed in table 4.1. In a multiplicity setup consisting of N identical detectors with efficiency ft, the probability F» that a cascade of M uncor- related gamma-rays causes the number of p detectors to fire (p- fold coincidence) is given by (Ha75) p P • M (4.8) Hp i-O In this expression it should be understood that a detector that is hit by more than one gamma-ray, belonging to the same cascade, only fires once. Expression (4.8) has been generalized to the case of nonidentical detectors by Van der Werf (We78). Various methods to 56 Eo(MeV) E*ax(MeV) (lab . energy) l5ONd(a,4n)15OSm 45 17.34 158Gd(a,4n)158Dy • 40 8.85 16OGd(a,4n)16ODy 40 10.88 16O 16O Gd(a,4n) Dy 49 19.66 Table 4.I.: Maximum attainable excitation energies for the va- rious reactions in column 1. extract information about the multiplicity distribution of gamma- rays from the measured k-fold spectra have been proposed in the li- terature. (Ha75,Ko78c,Oc78a,We78). This information is usually ex- pressed in terms of the average value W. = (4.9.b) Here p(M) is the normalized multiplicity distribution of the gamma- rays, i.e. I p(H) = 1 (4.10) M In the following sections an outline of the present model for the calculation of gamma-ray spectra following (H.I.,xn)-reactions will be given, and this model will be applied to a few (a,4n)-re- actions. Results will be compared to experimental data, obtained at the KVI by Ockels, de Voigt and others (Oc76,Oc77,Su775Oc78). 57 4.2. THE MODEL ••:• For the calculation of the gamma spectra and multiplicity dis- tributions of gamma-rays in (H.I.,xny)-reactions we use essentially the same model as for the neutron capture reactions. As was pointed out in section 3.6 an important difference between the two kinds of reactions is that, whereas in the case of neutron capture the ini- tial population has a sharply determined excitation energy, spin and parity, this is not the case for (H.I.,xnv)-reactions. Here we have to deal with a population distribution in excitation energy, spin and parity, so as to represent the entry states, from where the gamma deexcitation takes over from the neutron evaporation. The initial population should be calculated from a knowledge of the re- action mechanism. Conversely the population distribution can be es- timated by assuming some energy distribution for the evaporated neutrons, e.g. a Maxwellian distribution with a peak energy around 1 MeV. In the absence of a knowledge of the reaction mechanism, one has to assume a model population. As a way of illustration of our methods we consider here a particular initial population as follows. For the total population, present in the entry states with spin and parity J , we take C(2J+1) (£ ) (4.11) mm max Here C is an arbitrary constant. Expression (4.Ü1) can be derived in the bin-theory of Diamond et al. (Di76), which assumes that the angular momentum interval between £ = 0 and £ = 8 can be divided into a number of bins, each corresponding to a particular (xn)-re- action channel. In this picture the higher values of £ correspond to the lower values for x. Although the separation between the dif- ferent (xn)-channels in practical situations is probably not very sharp, (4.11) may represent a reasonable parametrization for the entry states if one interprets £ . and g as the effective mini- min max mum and maximum angular momentum respectively corresponding to a 58 * I particular number of evaporated neutrons. For the population dis- tribution in excitation energy we take a gaussian form centered around an excitation energy AE above the yrast line (See fig.4.2), i.e. 2 2 f(E,J) = 727-5 exp[-{E-E0(J)-AE} /2oE ] (4.12) where En(J) represents the excitation energy of the yrast state with spin J. 20 KVI1368 •max 16 12 E(MeV) 8 4 - 16 J 20 "Iman x Fig.4. 2.: Schematic representation of the parametrization of the populatiyn distribution, representing the entry statesa in terms max mm ^min* ^mox' °E' '^ attainable excitation ener- gy is denoted by the dashed line labelled E t max 59 The yrast line is parametrized by the following expression 2 3 4 EQ(J) = AJ+BJ +CJ +DJ (4.13) This latter parametrization is not essential and it can be replaced P(E,J,TT) = P(J,ir)f(E,J) (4.14) For the population distribution over states with different numbers of unpaired neutrons and unpaired protons we make the same statis- tical assumption as in (3.26), i.e. the total population distribu- tion representing the entry states reads p(E,J,7r,n,z) = P(E,J,ir) (4.15) x p(E,J,ir,n,z)/ Z p(E,J,ir,n',z') n\z' For the low-lying states, to be treated explicitly in the pre- sent calculation, we chose to take into account all the collective states below 2 MeV excitation energy, and in addition to these the first two excited positive parity states.for each J, as far as they are not contained already in the class of states below 2 MeV. For the reactions Gd(a,4n) Dy and Gd(a,4n) Dy we did not con- sider negative parity states because these do not seem to be popu- lated in the reaction. For the case of Nd(a,4n) Sm, in which reaction a large population of negative parity states is observed, we also took into account the first excited negative parity state of each spin. For these states and the electromagnetic transition rates between them we used the results of IBA calculations. (See Appendix A). It should be mentioned here, that, in the study of the deexcitation process following this reaction, we made no attempt 60 to take into account the influence of the GÜR in the way it was done for (n,y)-reactions. 4.3. RESULTS In this section the results of some calculations with the pre- sent model will be presented and compared to experimental data. The following (a,4n)-reactions will be discussed: Nd(a,4n) Sm (E, 45 MeV,Su77), 158Gd(a,4n)158Dy (E = 40 MeV,Oc77) and a a I6O 16O Gd(ct,4n) Dy (Ea = 40 MeV and Ea = 49 MeV,Oc77). The parameters used in the calculations are shown in Table 4.2. In order to avoid confusion we have given the spin cutoff factor a the index s. In s the same way as was extensively discussed in subsection 3.3.1, the pairing energy A was fitted in such a way that our level density formula (3.11) reproduces the total level density of Gilbert et al. (Gi65a,Gi65b) at the excitation energy of the neutron capture state. The middle part of Table 4.2 displays the values of A,B,C and D u- sed for the parametrization of the yrast line (eq.4.13). In the lower part the values used forfi . ,fi , AE and o_ (See fig.4.2) mm max E are shown. These four parameters are the crucial ones and contain most of the physical information of the process. In fact, as it will be shown below, they can be related to experimental results. For the purpose of showing how our method can be used in practice we have assumed some particular values. For the cases of 40 MeV and 49 MeV) Nd(ct,4n) Sm and Gd(a,4n) Dy we have chosen some reasonable the values of £ . ,£ and AE were taken from the work of Ockels estimates. For the case of Gd(a,4n) Dy (Ea et al. 'Oc77), whmion determinemax d AE by multiplying the experimentally observed average multiplicity and average gamma-ray energy of the side feeding of the ground state band levels. First we show in fig. 4.3 a comparison between calculated and measured (Su77) intensities of transitions between some discrete 15O 150 states in the reaction Nd(a,4n) Sm (Ea «= 45 MeV). The agree- ment between theory and experiment here appears to be good. 61 15ONd(a,4n) 158Gd(a,4n) 16OGd(a,4n)16°Dy I50Sm 58Dy E (MeV) 45 40 40 49 a 0 5.1* 4.8* 4.8* s P(N) (MeV) .99* .92* .73* P(Z) (MeV) 1.22* .92* .92* 1 gn (MeV" ) 4.0 4.0 4.0 g (MeV"1) 4.0 4.0 4.0 A (MeV) 1.0 .93 .87 A (MeV) 1.602 10"1 1.772 10"2 1.303 10~2 B (MeV) 9.005 10"3 1.680 10~2 1.463 10"2 C (MeV) -4.219 10~5 -4.327 10"4 -2.004 10"4 D (MeV) -5.949 10~6 5.876 10"6 6.348 10~7 e . (ft) 6 4 4f 6f min 20 15 17+ + 8ma x (ft) 19 AE (MeV) 9.5 3.5 4.0+ 7.1+ o (MeV) 4.0 2.5 3.0 4.0 * From Gi65a, Gi65b. From Oc77. Table 4.2.: Parameters, assumed for the calculation of the gam- ma spectra^ gamma-ray multvplioities and multiplicity distribu- tions of the various (a,4n)-reactions. 62 Since the calculated intensities depend strongly on the B(E1) and B(E2) values between the discrete states, the agreement between theory and experiment indicates that these are on the whole correct- ly given by the IBA-model. KVI 1369 ÜJ 45MeV Fig. 4. 3: Comparison between experimental (in parentheses) and cal- culated intensities of transitions within the ground state band3 the first negative parity band and transitions connecting both bands for the reaction Nd(a,34n) Sm (E = 45 MeV). The experi- mental data were taken from (Su77). 63 KVI I37Q l5O l5O 14 Nd(a,4n) Sm Fig.4.4: Comparison of cal- 12 oülated (dream curves) and experimental average multi- 10 - plicities t (Ea = 45 MeV). Experimental data were taken from (0a?8b). 0 0 4 8 !2 16 KVI 1371 l-i l58Gd(a,4n)l58Dy 10 Fig.4.5: Comparison of cal- Ea = 40 MeV culated (drawn curves) and experimental (Oc77) average multiplicitiess widths, a3 of gamma-rays in coincidence with discrete ground state band transiti- ons I+I-2 in Dy for the 4 - reaction 1S8Gd( 0 4 8 12 16 64 In fig. 4.4 the results of a calculation of the average ciulti- r -'. licities, V<(M- (4.16) Both the average multiplicities and widths, calculated with the as- sumed values of fi . , £niav , AE and c , are too low, but the trend mm max a in the experimental results is qualitatively reproduced. KVI 1372 l6OGd(a.4n)l6ODy 10 Ea=40MeV 8 4 2 0 8 12 16 I Fig. 4.6: Comparison of calculated (drawn curves) and experimental (0c77) average multiplicities, 14 Gd(a,4n) Dy Ea = 49MeV Fig.4.7.: Comparison of cal- 12 culated (drawn curves) and experimental (Oc77) average 10 multiplicitiest 12 16 Transition Energy(keV) I +c.e.b) I (theory) Y a) 98.94 36 ' 140 136 218.32 112 132 132* 320.62 100 105 120 6^ 406.14 78 80 100 10 8"* 475.9 54 55 74 529.3 29 29 45 14 12"* 563.3 19 19 20 578.0 11 11 1.3 4. | CO 1 CO Experinental data for Gd(a,4n) Dy Table 4.3.: Experimental and calculated intensities of ground 7 *5/? 7 *>/? state band transitions for the reaction Gd(as4n) Dy. 66 Transition Energy(keV) c.e.C> I (Theory) Y V E = 40 MeV E = 49 MeV a 0* 86.79 22a) N9a) 123 117 4*- 2* 197.02b) 107 133 120 115 6*- 4* 297.4 100 107 107* 107* 8*- 6* 386.0 77 79 95 98 10*- 8* 461.7 53 54 77 83 12*- 10* 522.8 32 32 54 63 14*- 12* 563.5 21 21 34 46 16*- 14* 576.7 11 11 9.2 20 18*- 16* 580.5 4.4 4.4 0.8 6.1 Experimental data for Gd(a,4n) Dy (E = 43 MeV, Jo72). * Normalized to experiment, a) \ No correction made for self-absorption in the target. b) The 197 keV peak contains also contributions from the 0( 16O 16O v Gd(a,4n) Dy (Ea = 40 MeV and Ea 49 MeV) comparisons of cal- culated and experimentally deduced (Oc77) average multiplicities and xvidths of multiplicity distributions of gamma-rays in coinci- dence with the I-I-2 ground state band gate transitions are shown in fig. 4.5, 4.6 and 4.7 respectively. In the case of 1 CO 1 CO Gd(ct,4n) Dy the experimental errors in reactions at 43 MeV bombarding energy in tables 4.3 and 4.4. 4.4. DISCUSSION OF THE RESULTS Although the parametrization of the population distribution representing the entry states, in terms of £ . , £ , AE and or (4.11 and 4.12), is very simple and does not include the details of the actual physical situation, it appears that the calculation can explain the trends that are experimentally observed in the a- verage values and the widths of the multiplicity distributions of gamma-rays in coincidence with certain discrete gate transitions. (See figs. 4.4, 4.5, 4.6 and 4.7). Moreover, the average multipli- cities, calculated with the assumed values of S. . , £ ,AE and Op, already agree with experimental results to withimm n maxabou' t 10%, for the reactions considered in the previous section. The above mentioned four parameters are related to observable physical quantities in the following way. The intensities of dis- crete transitions between high spin states in the ground state band are sensitive to the value of £ .A change of £ from 6 6Q max max 17h to 19l> for the reactions Gd(a,4n) Dy at 40 and 49 MeV bombarding energy respectively (Table 4.2) changes the calculated intensities for the 16 ->• 14 and the 18 -»• 16 transitions from 9.2 to 20 and from 0.8 to 6.1 respectively. (Table 4.4). The va- lue of £ . essentially determines the rate of decrease of the in- min tensities for the I-»-I-2 transitions with increasing I, in particu- lar for the values of I below £ . . The higher the value of £ . is. mm ° + mm ' latethe dlowe intensitier this rats efo ro f thdecrease 2 ->•e 0of 4intensitie -* 2 , 6s -is> 4.' (cfand. 8th'e -»•calcu 6' - 160, transitions for both *""Gd(a,4n) reaction+ -s+ in,+ tabl.e+ 4.4). „.+ The ,a+- verage value of the multiplicities, pre-equilibrium neutron emission may play a role in this case, gi- ving rise to higher energy neutrons. An experimental determination of the energy spectrum of the emitted neutrons could yield a decisive conclusion about the actual value of AE. Furthermore, it may be ar- gued that there is no a priori reason to assume that the centroid energies of the entry states constitute a line parallel to the yrast 158 line. In particular for the (a,4n)-reaction on Gd at 40 MeV the entry line is expected to be flatter in excitation energy. On the other hand there is experimental indication (Oc77) that at higher bombarding energy the entry states may rise even faster in excita- tion energy as a function of angular momentum than the underlying yrast states. So far only (a,4n)-reactions at relatively low bombarding e- nergies have been considered. The low-lying states that were taken into account explicitly in our calculations were always taken from a model calculation, using the Interacting Boson Model. This IBA- model without broken pairs has a natural cutoff value for the an- gular momentum 2(N (4.17) v N max IT Here Nv and N denote the number of neutron and proton bosons res- pectively. In (a,4n)-reactions at higher bombarding energies and in reactions initiated by heavier ions than a's, for which £ may be larger than J , one may need an extension of the IBA-model to include broken pairs. Moreover, at this high excitation energy, other alternative descriptions of the spectrum may be possible, like for example that suggested recently by Liotta and Sorensen (Li78) who took into account a number of parallel bands along the yrast line, described in terms of the wobbling model of Bohr and Mottelson (Bo75). 70 APPEINJDIX A t •:'• THE INTERACTING BOSON MODEL A.1. INTRODUCTION If one would know the location of all discrete states and the electromagnetic transition rates between them, these could be taken directly from experiment. However, with the exception of a few low-lying states, this information is not available. Therefo- re one has to turn to some model in order to calculate the gamma- ray transition probabilities between the discrete states, which go into the calculation of the deexcitation process. In this thesis the electromagnetic transition rates between the discrete states have been taken from the Interacting Boson Model of Arima and Iachello (Ar75,Ar76,Ar78a,b). This model is useful, because it gives simultaneously the electromagnetic trans- ition rates between and inside the various "collective" bands. Conversely one could have used another description, such as that of Bohr and Mottelson, as done by Liotta and Sorensen (Li78). In the Interacting Boson Model, to be referred to as IBA, collective states are described in terms of a system of N interac- ting bosons. These bosons have been later identified (Ar77,Ot78) with pairs of fennions coupled to L=0 (s-bosons) and L=2 (d-bosons) (See fig.A.l). However, for the purposes of the present work this identification is not essential, since the boson Hamiltonian will x x Fig.A.l.: The configuration s°cT in the boson F'u(O) model. 71 only be used as a way of providing a fit to the experimental data. f •:( The most general form of a one- and two-body boson Hamilto- nian in second quantized form reads e,Edd d m m m L=0,2,4 w 2) (O) (0) (0) 0) (0) • (l/^)vo[(dV) (ss) + (sV) (dd)< ] (2) (2) (0) (0 0) 0) • u2[(dV) (ds) ] * Ju0[(sV) >(Ss)< ]< t + Here d (d) and s (s) denote creation (annihilation) operators for the d- and s-bosons respectively. The parameters cT (L=0,2,4), Vj (L=0,2), u (L=0,2) are related to the two-body matrix elements by 2 2 cT = 2 2 2 The parentheses in (A.I) denote angular momentum couplings. If one wishes one could rewrite the Hamiltonian in terms of the 36 gene- rators of the group U(6), namely (s+s)(0\ (d+d)(0), (dfd)(l), + 2 + + 2 + (d d)< >, £j and the two-body matrix elements Cj, vL and u., the eigenvalues and eigenstates of H are found by diagonalizing (A.I) in a com- plete basis. The model contains as limiting cases some typical si- tuations, for which the eigenvalue problem can be solved analyti- cally. These situations occur when H can be written in terms of the generators of a subgroup G C U(C). The situation of an anhar- raonic vibrator is recovered if G H U(5) (Ar76), the rotational limit is obtained for G = U(3) (Ar78a) and the -y-unstable limit if G = 0(6) (Ar78b). Situations intermediate between vibrational and rotational limit can also very well be described (Sc78). Negative parity states are interpreted r\s arising from the interaction of a single octupole boson (f-boson) with the s and d boson excitations. The additional Hamiltonian, to be added to (A.I) for the negative parity states reads (Ar76) e Zf+f (L L £f m (dV) W >] m L=l,2,3,4,5 (A. 3) It is convenient, for purposes of diagonalization, to rewrite the Hamiltonian (A.I) in terms of some simple operators, namely the pairing term, P.P, the L.L-term, the quadrupole-quadrupole term, Q.Q, the octupole-octupole term, 0.0, and the hexadecupole- hexadecupole term. For simplicity, in fitting the positive parity states we omit the hexadecupole term, put E =0 and rewrite (A.I) s as H = e n + a p p + (A.4) o d d p - + aQQ.Q + aQ0.0 Here n, denotes the number operator for d-bosons. The single-boson energy e, in (A.I) is equal to the sum of el and the one-body con- tributions of the P.P, L.L, Q.Q and 0.0 terms in (A.4). We also 73 rewrite the additional Hamiltonian, to be added to (A.4) for the negative parity states as H' = efn£ + BQQd.Qf + (A. 5) The diagonalization was performed using the code PHINT (Sc76a). A.2. CALCULATED SPECTRA FOR 150Sm, 158Dy AND 160Dy In this section we present the results of the fit for the le- vel schemes of Sm, ' Dy, which were used in the calcula- tions of the gamma-ray spectra in Chapters III and IV. Exp Th Exp Th a> •'23 19- Exp Th in+Exp Th LÜ 150 Sm 150Sm Fig.A. 2.: Comparison of experimental and calculated (IBA) posi- tive parity (left) and negative parity (right) levels in Sm. In fig* A.2 we show the comparison between experimental and fitted positive and negative parity levels in ,0Sm00. The total OZ Bo number of bosons, N, which is equal to the sum of the active pro- 74 t % ton pairs, N = 6, and active neutron pairs, N = 3, is in this case N = N + N =9 (A.6) 7T V The parameters used in diagonalizing (A.A) and (A.5) are shown in Table A.I. 150Sm Parity + Parity - 384.55 keV = 1510. keV ap = -26.32 -2.50 „ aL = 1.00 „ = -19.96 „ -28.60 „ - 0. 0. „ Table A.I.: Parameters used for the calculation of positive and negative parity levels in Sm. In fig.A.3 and A.4 we show comparisons between observed and calculated levels belonging to the ground state band, the 3-band and the y-band in the nuclei Dy and Dy respectively. The experimental data were taken from Nuclear Data Sheets (Tu74a,b). The parameters obtained from the fit to experimental results are shown in Table A.2. In addition to the collective states, calculated using the IBA-model, non-collective (quasiparticle) states are expected to be present in the spectrum above ~ 1.5 MeV. The experimental oc- currence of these states, as compared with the calculated ones, can be seen in fig.A.5, where all known experimental levels below 2.0 MeV are shown in Sm. To the extent that these low-lying non-collective states have small matrix elements with the collec- 75 KVI 1374 g.s.b 20 158 Dy 18 16 > O) y LU \ 12 6 7 6 6 4 — — 5 8 2 — '4 o — —— 2 4 2 0 Exp. I.B.A Exp. I.B.A Exp. I.B.A Fig.A.3.: Comparison of experimental and calculated (IBA) levels 7 ^i*? belonging to the ground state band, the $- and y-band in Dy. tive ones, they can be altogether neglected from the discrete sub- space and only treated in the continuum part of the level spectrum. 76 KVI 1375 160 Dy 16 14 a> x2 12 P UJ r 1 i 10 — 6 —- — 2 — 5 4 — o o 8 V2 6 4 2 0 Exp. I.B.A Exp. I.B.A Exp. l.B.A Fig.A.4.: Comparison of experimental and calculated (IBA) levels 7 SO belonging to the ground state band3 the (3- and y-band in Dy. 158 160- I>y Dy -2.81 keV -60 .52 keV ap -44.26 ii -45 .67 „ aL 1.60 ii 3.45 „ -15.00 it -18 .50 „ aQ a o -2.83 ii -7 .75 „ Table -4.2.; Parameters used for the calculation of positive pari- ty levels in 158Dy and 160Dy. 77 KVI 1284 positive parity negative parity 2.0 1.5 — I x 1.0 Exp. I.B.A. 0.5 150 Sm 0.0 L 0 Exp. I.B.A Fig.A.S.: Observed positive parity (left) and negative parity (right) levels below 2.0 MeVt compared with theoretical results of the IBA-model, fo? 150Sm. 78 t%\ A.3. ELECTROMAGNETIC TRANSITION OPERATORS IN THE INTERACTING BOSON MODEL In a similar way as in section A.I, one can write down the most general form of a one-body E2 boson operator (A.7) which reduces to the quadrupole operator, 0, appearing in eq. (A.A) if q2=l and q^=-/7/2, i.e. 2) + + 2) + 2) Q< = (d s + s d>< - /7/2(d d)< (A.8) For the calculation of E2-transition probabilities in Sm we used for the 0.1,/a, = -1.051 (15°Sm) (A.9) while for Dy and Dy we took T^ proportional to the qua- drupole operator, i.e. 158 i6 q2/q2 = -fin ( ' °Dy) (A.10) The ilescription of the El-operator within the framework of purely collective models is more difficult. In first order, the El-operator can be written as Ti, = However, this operator fails to describe correctly the observed El-transitions, which are usually highly retarded. For this rea- son, higher-order terms may contribute appreciably, and one has (El) to resort to a much more complicated operator for T (Ar76, Ar78a,b). A reasonable fit to the experimentally determined (Su77) 79 B(E1)/B(E2) branching ratios in Sm is obtained by taking T"? with 3 Y1 = 3.28 io~ (A.13.a) 0.25 (A.13.b) In (A.12) n denotes the number of s-bosons: s s s (A.14) In our calculation we took for Sm the operator (A.12) with the parameters (A.13). From fig.4.3., where calculated and experimen- tal intensities of transitions within and between the ground state band and the first negative parity band are compared, it can be seen ehat the IBA-model describes the B(El)/B(E2) branching ra- tios as well as the bulk properties of the El- and E2-transition operators quite well. For Dy and Dy negative parity states were not considered and thus El-transitions between collective (discrete) states were neglected. 80 AFPEffflIX h THE COMPUTER CODE In this appendix we give a concise description of the orga- nisation of the various computer programs, used for the calcula- tions of the gamma-ray spectra and schematic flowcharts of the programs HIXNG and HIDISC. Table B.I gives an overview of the four main programs. The program PHINT (Sc76a) diagonalizes the boson Haid.ltonian (See Ap- pendix A). The eigenvectors are stored on disk for later use. The energies, spins and parities of the calculated collective states are written on a file separately. The program FBE2 (Sc77) calcu- lates the transition matrix elements in the Interacting Boson Mo- del, using the eigenvectors, calculated in PHINT. B(El) and B(E2) values are stored on the same file as the quantum numbers of the collective states. Next the program HIXNG (See also Table B.2) performs the cascade calculation of the statistical gamma-ray |f spectrum, either for the neutron capture process or for the (H.I., xny)-reaction, using the quantum numbers of the collective states, calculated in PHINT, as input. The statistical gamma spectrum, po- pulations and multiplicity distributions of gamma-rays are stored on disk. Finally the program HIDISC (See also Table B.3) calcula- tes the discrete gamma-ray transitions with the master equation technique, using the statistical populations, calculated in HIXNG as initial values. The quantum numbers of the collective states and B(E1) and B(E2) values from PHINT and FBE2, respecti- vely, serve as input for this program. The schematic flowchart of the program HIXNG is shown in Table B.2. The essential part of this program is the subroutine RLOIST, which calculates, given a population distribution over the level continuum, the redistribution of this population over the level continuum and the explicitly treated collective states, in the next cascade step. In actual calculations, discussed in Chap- 81 ters III and IV, 10 cascade steps turned out to be sufficient. The subroutine REDIST has options for El-, Ml- and E2-transitions in the statistical cascade. Hindrance factors for the various multi- polarities are read in with the general input. The subroutine FINDEL fits the value for the pairing energy, A, in the way as discussed extensively in subsection 3.3.1. The program HIDISC is schematically represented in Table B.3. The master equation method, discussed in Chapter II, is embodied in the subroutine DEEX, which at present can handle 300 states with transitions between any two of them. All calculations were performed on the CDC Cyber 74-16 com- puter of the Computer Center of the University of Groningen. 32 FHINT Diagonalizes Boson Hamiltonian E, J, TT, .... Eigenvectors B(E1), B(E2) FBE2 Calculates trans- ition matrix ele- ments HIXNG Cascade calcula- tion statistical spectrum Statistical spec- trum, multiplici- ties and popula- tions HIDISC Discrete transiti- ons, multiplicity distributions Table B. . 83 c HIXNG Main program r •''••] Read input FINDEL Fit A RD8 Read E,J,ÏÏ (from PHINT) POPIN Initial population (after neutron evaporation) REDIST Following cascade step GAMSPEC Print partial gamma spectrum WRSP Write statistical gamma-ray spec- trum, multiplicities and populati- ons to tape for use in HIDISC C Stop j Table B.2. 84 f HIDISC Main program ) 1 / Read / input 1/ » RD8 Read F.,J,ir,... 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To Prof. Dr. R.D. Lawson, who initiated me in the nuclear shell model, I largely owe my appreciation for the elegance and beauty of physics in general and nuclear physics in particular. Sincere thanks is due to Dr. R.A. Malfliet for the original, vivid and often entertaining way in which he taught me how compli- cated nuclear reactions can be described with relatively simple, classical methods. The results of a fruitful cooperation on the semi-classical theory of heavy ion scattering are to be found in a pair of publications (1,2). In a subsequent theoretical investigation of deeply-bound hole states in the odd tin isotopes (3), induced by experiments performed at the KVI, the theoretical input from Prof. Dr. F. Iachello and the many discussions with him, Prof. Dr. R.H. Siemssen, Dr. M.N. Harakeh and Dr. S.Y. van der Werf have been in- valuable. I thank my promotor, Prof. Dr. F. Iachello, for suggesting the topic of investigation for this thesis and I am most grateful to him and to my coreferent, Dr. A.E.L. Dieperink, for the many stimulating and enlightening discussions. The continuous interest in this work shown by Prof. Dr. R.H. Siemssen, Drs. W.J. Ockels, Dr. M.J.A. de Voigt, Prof. Dr. Z. Sujkowski, Dr. J. Lukasiak and Dr. A. Zglinski is gratefully acknowledged. I thank Dr. H. Gruppe- laar and Dr. J. Kopecky for the stimulating discussions that I had with them, in particular about the contents of Chapters II and III. The help of Mr. A.H. ter Brugge and Mr. L. Slatius in prepa- I- 91 ring the drawings has been instrumental. The help of Mr. W. Haai- ma in a last minute preparation of some of the figures is grate- fully acknowledged. I thank Drs. D.C.J.M. Hageman for carefully reading the ma- nuscript. '4 Finally, I want to thank my wife, Anneleen, for her patient • A and conscientious typing of the contents of this thesis and for her assistence in the preparation of the manuscript for printing. REFERENCES 1. T. Koeling and R.A. Malfliet, "Semi-classical Approximations to Heavy Ion Scattering based on the Feynman Path-integral For- malism", Phys. Rep. 22£ (1975) 181-213. 2. T. Koeling, "Semi-classical Approach to Heavy Ion Scattering based on the Feynman Path-integral Formalism", Proc. IV Int. Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg (1976) 162-164. 3. T. Koeling and F. Iachello, "Calculation of the Spreading- widths of the deep-lying Hole States in the odd Tin Isotopes", Nucl. Phys. A295 (1978) 45-60. 92 SAfENVATTINC Het doel van dit proefschrift is het geven van een beschrij- ving van het gamma-deëxcitatieproces van hoogaangeslagen toestan- den, gevormd in kernreacties. Omdat verschillende reacties zich slechts van elkaar onderscheiden ten aanzien van de manier waarop de hoogaangeslagen toestanden in het begin van de reactie worden bevolkt, kunnen de hier getoonde berekeningen algemeen worden toe- gepast op elke reactie, mits de beginpopulatie geschikt gekozen wordt. Het nieuwe aspect van deze berekening is de expliciete scheiding van de totale niveaudichtheid in twee delen: een discreet gedeelte, dat expliciet wordt behandeld, en een continu gedeelte, dat statistisch wordt behandeld. Nog een belangrijk nieuw aspect in dit statistische gedeelte van de totale niveaudichtheid is, dat, afgezien van een classificatie van de toestanden naar excitatie- energie (E), spin (J) en pariteit (ir), het aantal ongepaarde neu- tronen (n) en protonen (z) expliciet wordt ingevoerd. Bij het be- rekenen van electroraagnetische overgangen tussen toestanden hebben we dus te maken met drie soorten overgangen: (i) continuüm-conti- nuüm; (ii) continuüm-discreet en (iii) discreet-discreet. Het op- treden van een selectieregel voor de electromagnetische overgan- gen tussen toestanden in het continuüm, ingevoerd door de quantum- getallen n en z, speelt een zeer belangrijke rol in het bepalen van de eigenschappen van het deëxcitatieproces. Deze selectieregel weerspiegelt het één-deeltjes karakter van de electromagnetische overgangsoperator en houdt in dat de quantumgetallen n en z met niet meer dan twee eenheden kunnen veranderen in elke gegeven over- gang. Nog een ander belangrijk verschil tussen deze berekening en andere is de manier waarop de discrete toestanden worden behandeld. Hier wordt dit gedaan met behulp van een model, dat niet alleen een goede beschrijving geeft van de bekende discrete toestanden, maar dat het ook, in goede benadering, mogelijk maakt alle andere (experimenteel onbekende) collectieve toestanden in aanmerking te nemen. Omdat deze toestanden grote matrixelementen hebben met de 93 laagliggende toestanden, spelen ze een belangrijke rol in het de- excitatie proces. In dit proefschrift worden twee verschillende reacties bestu- deerd, namelijk laagenergetische neutronvangstreacties en (H.I.,xny)-reacties. Zoals boven reeds vermeld, is het verschil tussen deze beide processen voornamelijk gelegen in de beginpopula- tie van de toestanden: in het eerste geval heeft de beginpopulatie, die het neutronvangstniveau representeert, een scherp bepaalde ex- citatieenergie, spin en pariteit, terwijl we in het geval van (H.I.,xny)-reacties te maken hebben met een populatieverdeling, zo- wel in spin als in excitatieenergie, die de zogenaamde entry-toe- standen karakteriseert, die overblijven nadat de neutronenemissie heeft plaatsgevonden. We beginnen in Hoofdstuk II met aan te tonen dat het deëxci- tatieproces van een verzameling aangeslagen kernniveaux door middel van het uitzenden van gammastralen, equivalent is met een master- equation-systeem. Na een wat algemenere inleiding tot master-equa- tion-systemen wordt er een master-equation-techniek voor de bere- kening van gamma-spectra en de analyse van voedingstijden in kern- reacties ingevoerd. Deze techniek wordt dan, in combinatie net een cascade-techniek, toegepast op (n,y)- en (H.I.,xny)-reacties in de rest van dit proefschrift. In Hoofdstuk III worden neutronvangstreacties in detail be- schreven. Een overzicht van hieraan voorafgaand werk is te vinden in §3.1. Details van de onderhavige methode worden hierna uitge- breid besproken. In het bijzonder wordt er uitgebreid aandacht be- steed aan de niveaudichtheidsformule, die gebruikt wordt voor het beschrijven van het hoogliggende continuüm van toestanden, aan de beginpopulatie, die de neutronvangsttoestand representeert, en aan electromagnetische overgangswaarschijnlijkheden. De invloed van de reuzen-dipoolresonantie op El-overgangswaarschijnlijkheden wordt besproken in §3.3.3. De resultaten van berekeningen voor het geval 149 van thermische neutronvangst in Sm worden vergeleken met expe- rimentele gegevens. Verder wordt er aangetoond dat een experimen- 94 tele bepaling van multipliciteitsverdelingen van gammastralen in coïncidentie met bepaalde discrete overgangen gebruikt kan worden als een middel om belangrijke informatie te verkrijgen over de irultipolariteiten van gammastralen in de statistische gammacascade die het vangstniveau deëxciteert. In Hoofdstuk IV worden (H.I.,xny)-reacties besproken. De pro- cedure wordt daar toegepast op de reacties Nd(a,4n) Sm, 158Gd(a,4n)158Dy en I6OGd(a,4n)16ODy. De resultaten van de bere- keningen worden vergeleken met experimentele gegevens, verkregen aan het KVI door Ockels e.a. (Oc78). Aangezien we voor de beschrijving van de laagliggende (col- lectieve) toestanden en de electromagnetische overgangswaarschijn- lijkheden daartussen gebruik maken van het IBA-model van Arima en Iachello (Ar75,Ar76,Ar78a,b), wordt er in Appendix A een korte schets gegeven van dit model en worden de berekende niveauspectra voor d,e kerne, n 150Sm„ , 158D„y en 160D„y gegeven. Tenslotte wordt in Appendix B in het kort de organisatie en opbouw beschreven van de verschillende computerprogramma's, die bij de berekeningen werden gebruikt.