The Measurement of Static Electric Quadrupole Moments of Excited Nuclear States by Coulomb Excitation Citation Stokstad, Robert Grandin (1967) The Measurement of Static Electric Quadrupole Moments of Excited Nuclear States by Coulomb Excitation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V414-DF47. https://resolver.caltech.edu/CaltechETD:etd-10072002-122527 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Measurements of the probabilities for Coulomb excitation of the first [...] excited states in [...] and [...] [...] have been made with [...], [...] and [...] ions. The deviations observed in a comparison of the experimental data with the predictions of first order perturbation theory are interpreted in terms of the static electric quadrupole moments, [...], of the first [...] excited states of the nuclei. The deduced values of [...] are found to lie in the ranges [...] and [...] for [...] and [...] [...], respectively. The deduced values of [...] for [...] and [...] are found to be surprisingly large when compared with the predictions of the vibrational model. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Barnes, Charles A. Thesis Committee: Unknown, Unknown Defense Date: 6 December 1966 Record Number: CaltechETD:etd-10072002-122527 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10072002-122527 DOI: 10.7907/V414-DF47 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3951 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 07 Oct 2002 Last Modified: 20 Mar 2024 21:14 Thesis Files Preview PDF (Stokstad_r_1967.pdf) - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

f'HE MEASUREMENT OF STATIC ELECTRIC QUADRUPOLE MOMENTS OF EXCITED NUCLEAR STATES BY COULOMB EXCITATION Thesis by Robert Grandin Stokstad In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology | Pasadena, California 1967 ‘(Submitted December 6 , 1966) ii ACKNOWLEDGMENTS The experiments reported here were carried out with the collaboration at various times of Dr. Jorrit de Boer, Dr. Jan Hall, Dr. Geoffry Symons and Dr. Aage Winther. The author wishes to thank these people for the opportunity to work with them. Special thanks are due to Dr, de Boer, who suggested the experiment and whose constant interest and participation in these measurements are greatly appreciated. . Professor Charles Barnes has maintained an interest and concern in the author's progress as a student beginning with the author's first days at Caltech, The author wishes to especially thank him for his guidance and encouragement, Financial assistance was provided by the California Institute of Technology, General Atomic Corporation and Rand Corporation and is gratefully acknowledged. Finally, thanks are due to the entire staff and personnel of the Kellogg Radiation Laboratory for creating the stimulating and pleasant atmosphere which have made the author's stay at Cal Tech a memorable experience. ili ABSTRACT Measurements of the probabilities for Coulomb excitation of the first 2* excited statesin “44Cd, “4eCd, *2oTe and +2876 have been made with 4 He, 12. and 166 ions. The deviations observed in a comparison of the experimental data with the predictions of first order perturbation theory are interpreted in terms of the static electric quadrupole moments, Q,, of the first 2” excited states of the nuclei, The deduced values of Qo are found to lie in the ranges -0. 44 > Q, > -0. 97, -0.67 > Q 21 18, -0.16 > Q, = -0.50 and 20.01 = Q, > -0,40e+ 10°24em2 for 14¢q d, Mca, 12676 ana 128 2 08 tO tor tnt ang t8¢ | BQ Te, respectively. The deduced values of Qo for 48 Cd an 48 Cd are found to be surprisingly large when compared with the “reticle ne of the vibrational model. | iv TABLE OF CONTENTS PART TITLE PAGE I INTRODUCTION pal THEORY OF COULOMB EXCITATION A. Fandamental Assumptions The Semiclassical Approximation Method of Solution (1) First order perturbation theory B, C, Multipole Interactions dD. oO © ab oH ft & Mt (2) Second order perturbation theory 11 (3) Symmetrization of first order cross sections 17 (4) Numerical integration of the coupled differential equations 18 E. Summary 19 Til EXPERIMENTAL APPARATUS AND METHOD . 20 A. Heavy Ion Beam Production 20 B. Target Preparation 20 C. Detection System al D. Procedure 23 E, Auxiliary Experiments 24 IV -—s ANALYSIS AND RESULTS , | 26 A. Corrections Applied to the Raw Experimental Data 26 © B. Presentation of the Data , 29 C, Higher Order Effects 29 D. Least-Squares Fit to the Data 33 E. Results — 35 Vv. DISCUSSION OF THE ANALYSIS 38 VL DISCUSSION OF THE RESULTS 42 PART TITLE. APPENDIX Tr SYNOPSIS OF THE MULTIPLE COULOMB EXCITATION COMPUTER PROGRAM ey AUXILIARY MEASUREMENTS A. Magnetic Analysis Measurement B. Gamma-gamma Coincidence Measurement C, Sulphur Beam Measurement: mt GAMMA RAY SPECTROMETRY REFERENCES TABLES FIGURES PAGE 54 59 59 62 64 67 71 74 82 I. INTRODUCTION The Coulomb excitation process is a powerful technique for the investigation of the structural properties of the nucleus, Its particular advantages arise from the fact that the energy of the bombarding particle is held below the Coulomb barrier. The Coulomb repulsion prevents the surface of the projectile from penetrating that of the target nucleus, thus restricting the inter- action between the two nuclei to the well-understood electrical force and enabling an unambiguous, model-independent measure- ment of certain nuclear parameters, Coulomb excitation has been used extensively to measure the energies, spins, parities and E2 radiative transition matrix elements of low-lying nuclear states (Alder, et al., 1956). The ensuing wealth of systematic information made available has been a major contributor to the development of the collective model of the nucleus, As experimental equipment and techniques improve, Coulomb excitation continues to provide the nuclear theorist with the additional information he needs to test and improve his models. This thesis describes a series of experiments in which Coulomb excitation has been employed to measure the static electric quadrupole moments of first-excited 2° states in the even- even nuclei 11464 11664. 12676 and 12856, In Section I the nature and theory of the Coulomb excitation process are reviewed with emphasis on those aspects relevant to the design of these experiments. The experimental apparatus and method, and the analysis and results are presented in Sections I and IV, respectively. A discussion of the assumptions and approximations used in the analysis is given in Section V. In Section VI the results are discussed in the light of existing nuclear models and are shown to represent a challenging problem for the theorist, 3 i. THE THEORY OF COULOMB EXCITATION A. Fundamental Assumptions Coulomb excitation refers to the production of nuclear excitations by a long-range electric interaction with a bombarding . particle. In particular, when the energy of the bombarding particle is well below the top of the Coulomb barrier!) ; Eo) the interaction between the projectile and target nucleus will be free from interference by any nuclear forces. There are, however, two additional conditions which must be satisfied before one may proceed to the assumption of a classical orbit for the projectile (Alder, et al., 1956), These conditions are: (i) The de Broglie wavelength, 4, of the projectile must be much less than the distance of closest:‘approach, 2a, ina head-on collision, This requirement may be expressed as where Z4 and Zo are the respective nuclear charge numbers of the projectile and target, and v is the initial relative velocity of the projectile. One half the distance of closest approach in a head-on 1) If one takes r = 1, 44 avs, 10° }3om for the nuclear radius, the formula for the Coulomb barrier energy, EW has the 2145 simple numerical form, EB, = (MeV). Ay? a,¥8 collision is given by 2 44 Ao as 3. mV where m O is the reduced mass. (ii) The energy lost by the projectile in producing a nuclear excitation may constitute only a small fraction of its initial energy. Conditions (i) and (ii) are satisfied most easily by using heavy projectiles, e.g., tHe, 12¢ or 166 ions to excite low-lying ( < 1 MeV) collective states. This is illustrated in the following table in which the values of E, and 7 are given for various projectiles incident on Maca and with energy equal to E, (in MeV). projectile E, ia e* 9.9 0. 35 p 8.2 2.6 4 ‘He 14,9 7.8 16, 52.1 34, 0 The energy of the first excited state in /44ca is 0.56 MeV. B. The Semiclassical Approximation A classical treatment of the projectile orbit produces the Rutherford differential cross section | do, role) 2 sin'* (2) (1) ALR » n 5 which describes the angular distribution of the scattered projectiles. The differential cross section”) for the Coulomb excitation of a state f from an initial state i may be written as . do do _ R da * Pine WH (2) where P. f is called the probability for Coulomb excitation. P.. f is given in terms of the transition amplitudes, b.,, by Pop ey ) lbel? (3) i M,, M, ? where I is the spin of the initial state of the target nucleus and M. and M, denote the respective magnetic substates of the initial - and final states. The central problem of Coulomb excitation theory is thus the calculation of the transition probability amplitudes Die 2) Hereafter, ''cross section" is understood to mean the differential cross section, The amplitudes bs are obtained from the solution of the time-dependent Schrodinger equation . Hy = in SY , (4) Since the electric field of the projectile may be regarded as a perturbation localized in time and space, one may write H= Hy + Hat (t) (5) and Lge | y=) at)|nye” 7. (6) n H O is the Hamiltonian of the free nucleus with eigenfunctions Hee) is the energy of the charge distribution of the nucleus in the known, time-dependent electric field of the projectile. One thus obtains the set of coupled differential equations ig -E_ )t ihaw=) GlB,,@lme" 2 ™ ale (7) m for the amplitudes a, of the free nucleus eigenfunctions Wa It is noted that b,, is equal to a(t ~ o} for a nucleus initially in state i, i.c., with a, (t +-o) = 4, The classical treatment of the projectile orbit resulting in (1), together with the quantum mechanical treatment, (4), (5) and (6), of the nucleus in the time-dependent electric field, produces (7) as the equation for the amplitudes of the free nucleus eigenfunctions. Equation (7) is therefore referred to as the | "semiclassical approximation". C. Multipole Interactions The interaction Hamiltonian, Ht) , is separated into its multipole components, of which there are four types (Alder and Winther, 1966): (i) the monopole - monopole interaction. This is simply the interaction which produces the Rutherford scattering and is subtracted out in subsequent calculation. (ii) the monopole - multipole interaction. The multipole moment of the target nucleus interacts with the point charge field of the projectile to cause an excitation. (iii) the multipole - monopole interaction, Excitation of the projectile proceeds through this mechanism. Since (ii) and (iii) are incoherent, the latter need not be of concern here. (iv) the multipole - multipole interaction. This is generally negligible, Of the various monopole - multipole interactions, denoted by EX and Ma, 4=1,2,3,..., the electric quadrupole interaction (E2) has been observed to occur most frequently in the Coulomb 3) character of the collective oscillations and deformations prevalent excitation process This is a consequence of the quadrupole in nuclear matter, and also of the selection rules to which the Coulomb excitation process is subject. For the case of E2 multipole interaction only, Ht) becomes 2 4 -3 H(t) == 24°). Pp () ¥p 0.0, o()) E24) 0) where 2 ~\ 43 M(E2,u) = | ro Vy | (@,o)0,(r) dr (9) and Ty By and Py denote the coordinates of the projectile relative to the nuclear mass center. p(T) is the nuclear charge density; ”(E2,u) is referred to as the E2 multipole moment operator of the nucleus. 3) A few cases of El excitation are known, notably in Oy and E3 excitations have also been observed (Litherland, et al., 1963, and McGowan, et al,, 1965). The relevance of a virtual E1 excitation to these experiments will be discussed in Section V. Magnetic multipole excitations are inhibited by a factor (v/c)2 and are further reduced for large angle scattering. D. Methods of Solution There are several methods which may be employed to solve (7) and thereby obtain the excitation probability. The approximations used in these various methods will serve to classify the various types or modes of Coulomb excitation. (1) First order perturbation theory The excitation probability is assumed small such that the amplitude a. of the ground state remains constant and equal to unity with all other ae << 1. In this case, ee, i (fH [ide ao Et p(t) f if dt . (10) sa,ek f ia A change of variable in (10) followed by integration over the orbit of the projectile yields (Alder, et al., 1956) df. (8, Se.) | (1,1). 2 pel Se) 4g | Pit “Xia aigu 0) Sg) (11) where -14,36A 2/2 23/2 Xt" +s 3 (12) (14.4 ,/Ag)°Z,Z,"(21, +1) Zy Zp,” AE g.. = (13) f 72? 7 12.655 10 ABS, = (Ep- ED) (+ Ay/Ay) , | ay and M,, = (4l]i* m(Ea)|1],), he2. (15) E is the initial laboratory energy in MeV of the projectile with charge 2, and mass Ay ina,m,u. 6 is the center-of-mass angle through which the projectile is scattered by the target nucleus (Z, Ad) and E, and E, are the energies in MeV of the final and initial nuclear states, respectively. The orbital integrals dopo(6, 5) have been tabulated by Alder, et al., (1956); the parameter a is the ratio of the collision time (a/v) to the nuclear period (4#/AE) and indicates the extent to which the excitation process proceeds adiabatically. The reduced matrix element M,. connecting the initial and final states is in units of e- 10°2* em”, Xqug is a measure of the strength with which the final state is coupled to the initial state through the interaction of the E2 multipole transition moment of the nucleus with the monopole field 4) er is the excitation probability in of the bombarding particle a sudden, i.e., § fi Q, head-on collision. The assumption of perturbation theory may therefore be stated as Xe << 1. It is noted that M2 = B(E2, s~ r) (21, + 1) : (16) r 4) Xp is analogous to the parameter n , which measures the strength of the monopole - monopole interaction, 11 where B(E2) is the same nuclear matrix element appearing in the expression for the E2 radiative decay of the state s to the state r, As a consequence of this, the Coulomb excitation process is subject to the same selection rules which apply to the radiative decay of the nucleus. (2) Second order perturbation theory The amplitude a, of an intermediate siate z is included in (7) to produce two coupled differential equations. The transition amplitude bie in second order perturbation theory (Alder, etal., 1956) then becomes: pl?) = pl) + » ») (17) where r ) . i(E - BY) 2 1 ar we bso "Z?! dt (£)/H) | ze , - iE, - E, ) t 2h x| dt’ (2|H)]ide So. (18) The summation in z is to be performed over all intermediate states, including the initial and final states. Insertion of (17) into (3) yields the perturbation expansion P= BO) 4 es p(22) ingt * “jeg-f * vA 12 The terms p(t2) and p(22) describe the phenomenon of multiple Coulomb excitation. For example, pi?) + ot is the probability of the projectile exciting a state of spin nd arity 4° froma 0° ground state by two successive E2 transitions. To illustrate a situation which corresponds to the actual measurements, one considers the excitation of a nucleus with or ground state and two excited ar states, which are labeled with the indices i, f, and z, respectively (see Figure 1). The Coulomb excitation of the first 2” state can then proceed by direct excitation from the ground state (first order, i- f) or by double excitation with either of the 2” states as the inter- mediate state (second order, i-z-f or i-f-f). Neglecting terms higher than third power in Xo? the probability of ex- citation of the first 2* state is = pill), p(12) (12) Pot = Prog * Pree * Pee - (19) The first term pa) (proportional to a) is the first order excitation probability defined by (11). The other two terms (proportional to x ¢) arise from the interference between first and second order excitation of the state f where, in the second order excitation, the states z and f are the respective inter- mediate states. The term pu) gives rise to a dependence of 13 the excitation probability on the static electric quadrupole 5 moment of the 2* state (the so-called "reorientation effect" ~’). The second order terms p(22) in Figure 2 are proportional to Kg and are of the same order as the terms arising from the interference between first and third order excitation. Calculations to this order have been made by Masso and Lin (1965). The probability pi) in (19) is given by 3/2 M. (12) _ 270 3 zi fz fi The quantity J (Coe Fe? 9) is related to the orbital integrals if 6,5) and By ho5 450, 9) defined by Alder, etal, (1956), by the relation 5) The term "reorientation effect'' was introduced about ten years ago by Breit and Lazarus (1955) who were the first authors to suggest the possibility of measuring a static quadrupole moment by Coulomb excitation. Recently, "re- orientation effect" has been used as a general expression to cover any effects of the static quadrupole moment in Coulomb excitation. Strictly speaking, however, it should be applied (Breit and Lazarus, 1955) only to virtual transitions among, and a redistribution of the final populations of the magnetic substates of the final state. This "reorientation" of the spin of the final state may be manifest, for example, in the angular distribution of de-excitationgammarays. However, for 180 degree scattering such virtual transitions and redistributions are excluded and only the m = 0 substate is populated. On the other hand, the effect of the static quadrupole moment on the magnitude of the cross section is a maximum for 180° scattering, In the experiments described here it is this latter property which has been exploited, 14 = _4\K TT Core 2? 8) = fr C1) You & 1 OF | (0, 555 +550) Po. 225, ee 0) « (21) Tables of the quantities Bien have been computed for positive §- values by A. C, Douglas (1962). The term pi?) | is obtained from (20) by setting z equal to f. The largest contribution to P. af arises from pi (11) which is proportional to B(B2,i-f), The term p{t2) ), which is of the order of magnitude of x is proportional to B(E2,i-f) and to the static quadrupole moment of the state f. In order to obtain an estimate of the size of this term relative to the excitation probability of the first 2° state it is convenient to consider the quantity p(t2)_; p(11) ea), Pitot i-f ° By substituting (11) and (20) into (22), one finds -4,6 int M ad ee sin “(3 p=1.68 y. Writing x, , in terms of &,, using (12), (13) and (14), one may rewrite (23) in the form o=- 2 +5 AB Mgy x K(G,, 0) (24) Z 2 2 (1+ A,/Ag) 15 where IE p40, 8) Sey Ha (8, 5e)/d0 sin * (2) . (25) K(é fe 6) = 1.90 5 The dependence of KE, 8) on Cri and 6 is illustrated in Figure 2. ? The static quadrupole moment Qe of the state f is related to the reduced matrix element Mee by Q.= -0,758 M ep = +0. 758 (I, || m(E2) || 1.) . (26) From (24) and Figure 2, it can be seen that: 1) the ratio p is directly proportional to M,,, and thus to the static quadrupole moment of the state f; a determination of the sign as well as the magnitude of Qo is therefore possible. 2) p is roughly independent of § gy and therefore of the energy of the projectile; 3) p is roughly proportional to A,, the mass of the projectile; 4) » is largest at backward scattering angles. In the present experiments, therefore, one compares the excitation probabilities for various projectiles scattered at anglics closc to 180°. 16 Since pill) and p(t?) ¢ 2re both proportional to (i, ,)*, the ratio of the excitation probabilities for different projectiles is independent of B(E2, o* - 2*) , leaving Moo as the only nuclear parameter in the ratio. Similarly, if the data are obtained under the same experimental conditions, many factors important to the determination of the absolute excitation probability will tend to cancel in the ratio. Notably, this removes the uncertainty in the efficiency of the NaI detector (+ 8% ) from the determination of the quadrupole moment, By way of numerical illustration of the above procedure, the values of PY and p are 0.175- (M,,)* and -0.181+ Myo, respectively for 40 MeV 16, ions scattered from orl at 165° in the laboratory system. The corresponding values for 10 MeV tHe bombardment are 0. 0163 - (Mt, 4)” ant 036 - Myo. if the term P.O’; im (19) is small relative to Peg? the excitation probability for the first 2" state may be written as = pit) Pig = Pug G+) In this case, the ratio of the excitation probabilities obtained for bombardment by 40 MeV 165 and 10 MeV 4He ions is given by Pi (°0) sore 1 18t Mop 17 Implicit in this treatment is the assumption that p{t2) dominates pit) ¢ and other higher order terms which are not included in the expansion (19), This condition is found to be satisfied in these experiments. (3) Symmetrization of first order cross sections It is shown by Alder, et al., (1956) that the use of symmetrized expressions for the adiabaticity parameter § fi and the distance of closest approach 2a brings the classical first order cross sections into closer agreement with the more accurate quantum mechanical calculations. The symmetrized expressions. for rH and do/dQ are 1/2 7. ZA _ 412084 ry-1/2 1/2 St —@gap 7 LM - AE.) "- EB] (27) and (11) ' do AE,. do E2 _ (1,0) _ fi R = PEP Se) G- a) ae (28) dQ inf where §,. is given by (27), and do, /da is the (unsymmetrized) Rutherford cross section. These symmetrized expressions for & (11) and for the first order cross section do are used throughout - the remainder of this thesis. 18 (4) Numerical integration of the coupled differential equations A perturbation theory treatment of the excitation process loses validity rapidly as the size of Xpof increases, Figure 3 shows for 11404 the range of values of Xp 25 a function - of projectile and bombarding energy” . For large Xq og? either the slowly converging perturbation expansion must be extended to higher orders in Xp OF a different method of calculation must be employed to account for the presence of non-negligible ampli- tudes a, in (7). The latter approach has been pursued by A. Winther and J, de Boer (1965). These authors have written a computer program which computes the cross sections for electric quadrupole excitation in the semiclassical approximation by numerically integrating the time-dependent Schrodinger equation (7) for a system consisting of a nucleus with a finite number of states and a projectile moving on a classical orbit. In addition to the bombarding conditions, spins and energies of the target nucleus levels, the program requires as input data the matrix Mig between all nuclear states considered in the calculation. A brief. description of this computer program is given in Appendix L By varying the values of the reduced E2 matrix elements M_. in the input to the computer program, one can study the rs dependence of the excitation cross sections on the different nuclear 6) In the experiments Xi. Was typically 0. 3 for 16, bombardment of 144, One experimental datum was taken with 46 MeV 16, ions; Xe in this case was about 0.5. 19 matrix elements for various bombarding conditions. Since, in the present experiments, it is found that this dependence is roughly approximated by a perturbation analysis, it will be convenient to retain the language of perturbation theory in discussing the results of-the computer calculations. For example, in discussing the . effects of a change in sign of any of the matrix elements in (20) one may speak of the sign of the interference term involving these matrix elements, indicating that these effects are in fact roughly « M M s fi zi ‘fz In the analysis the computed cross sections have been proportional to the product M fitted to the experimental data in order to obtain values for B(E2, or a*) and for the static electric quadrupole moment, Qo, of the first excited state. E, Summary It has been seen that the Coulomb excitation process - provides a means of investigating certain features of nuclear structure, notably the nuclear transition matrix elements B(E2). It was further shown that an extension of Coulomb excitation theory beyond that of a first order approximation enables a determination of static quadrupole moments. It is interesting to note at this point two significant features of these measurements: (i) nowhere in the development of the theory has any reference been made to a nuclear model, and (ii) the deduction of Q» from the measured cross sections does not require a separate estimation of an electronic electric field gradient at the nuclear site, as would be the case in a measurement of hyperfine structure, The electric field gradient here is provided by the bombarding particle, whose orbit is assumed to be known. ‘20 I. EXPERIMENTAL APPARATUS AND METHOD The experimental quantities to be measured were the probabilities for Coulomb excitation of the first 2” states of 144 116, d 126, 128 ; e and viz. , tHe, 124 and 16) These probabilities were measured by ? Te when bombarded with various heavy ions, counting the de-excitation gamma rays in coincidence with the scattered projectiles. A. Heavy Ion Beam Production The experiments described here were carried out with particle beams from the ONR-CIT tandem accelerator. The He ion beam was produced by the injection into the tandem of a 700 12, or 16, were keV neutral 40 beam, Negative ion beams of obtained by charge exchange of positive ions from the duoplasmatron source, using as source gases methane and a mixture of 5 to 10% of oxygen in hydrogen, respectively. B, Target Preparation Separated isotopes in the form of CdO and Te were obtained from the Oak Ridge National Laboratory. The isotope enrichments varied upward from 97% with the remaining constitution of the sample known, The targets were prepared by the vacuum evaporation of the separated isotope onto thin carbon foils. A light-weight material such as carbon was used for the target backing in order to produce a wide separation in the energy of the tHe ions scattered from the target material from those scattered from the backing. 21 The thicknesses of the targets were measured by recording the momentum profile of 1 MeV protons scattered at an angle of 140°, A second method also used was to compare the cross sections for elastic scattering of low energy (about 4 MeV) Sue ions from the target and from a nickel foil of known thickness, Values which were estimated to be accurate to + 20% were thus obtained for the energy loss of the projectile in the target. . Since the coincidence measurements do not require the resolution of the inelastically scattered ions from the elastically scattered ions, thicknesses corresponding to an energy loss of up to about 1.5 MeV for 165 ions could be employed. The thicknesses of the various cadmium targets ranged from 250 to 450 ugm/ cm’, After the cadmium measurements were completed, it was realized that thinner targets were preferable because of a resulting lower random- coincidence count rate and count-rate loss, Furthermore, since the probability for Coulomb excitation is strongly energy dependent, thinner targets produced a smaller error in the correction applied for the energy loss of the bombarding particle in the target. For the tellurium measurements, therefore, targets of thickness about 80 uy gm/ com” were prepared, C. Detection System . The arrangement of the detectors is illustrated in Figure 4, The heavy ion beam was confined to an area of 1.5 mm diameter by a series of tantalum collimators before passing through the center- hole of an annular solid-state detector. The target was placed about one centimeter from the sensitive area of the counter which then accepted particles scattered between laboratory angles of 155° and 22 175°, than sufficient to stop particles of all energies used in the experi- The depletion depth of the detector (about 300 uw) was more ments. The gamma detector was a 7.62 - cm - long by 7. 62 - cm - diameter Nal(T1) crystal mounted at an angle of 58 degrees with respect to the incoming beam, with its face about 3 cm from the target center. This angle is near the zero of the Legendre poly- nomial Py, which helps to minimize the effect of the anisotropic gamma-ray angular distribution, For this particular geometry the photopeak efficiency of the NaI detector is about 5% for gamma-ray energies near 500 keV. The coincidence circuitry consisted of a conventional fast- slow coincidence system, with the fast coincidence unit modified to provide a time-to-pulse-height conversion. A block diagram of the electronic apparatus is given in Figure 5. A 400-channel analyzer was gated by the slow coincidence unit which could be arranged to require a time coincidence of some or all signal inputs from (i) the fast coincidence unit; (ii) a single-channel analyzer which selected pulses from the gamma-ray spectrum corresponding to the photo- peak of the Q* 07 transition; (iii) a single-channel analyzer which selected pulses from the particle spectrum corresponding to scattering from cadmium (or tellurium). 23 D. Procedure — Figures 6 and 7 show for 11404 and 128.76 respectively, singles gamma-ray spectra and spectra gated by the requirements (i) and (iii) above. Corresponding singles spectra of 16, ions are shown in Figures 8 and 9, The relative thicknesses of the cadmium and tellurium targets are apparent from the widths of the peaks in these spectra. The single-channel analyzer level or "window" settings are also indicated in Figures 6 through 9. The fast coincidence unit gave an output pulse proportional to the time overlap of 90-ns-long clipped pulses derived from the particle and gamma detectors. A time-to-pulse-height spectrum, gated by a coincidence of signals from (ii) and (iii) above, is shown in Figure 10. Delays were adjusted such that pulses in real coincidence had about half-maximum time overlap to facilitate the subtraction of the uniform random-coincidence background. This subtraction was typically about 2%, never more than 5%, and was known accurately enough so that negligible error was incurred. The shape of the time-to-pulse-height spectrum was independent of the bombarding particle. After the subtraction of the random-coincidence background in the time-to-pulse-height spectrum, the experimental datum is the ratio, R, of the number of counts in the time-to-pulse-height peak to the number of particles (counted by a single-channel analyzer and scaler) scattered from cadmium or tellurium, Thus . n =f. do; j=2 “i Re -y (29) z v1 2% 24 where do, is the cross section for excitation of the jth level and n is the number of levels significantly populated in the experiment. Figures 11 and 12 show the energies, spins and labelling indices, i, of the levels considered in the analysis. The quantity f. is the probability that, after Coulomb excitation of the state i, the sub- sequent gamma decay produces a pulse in the window of the gamma- ray spectrum. For i= 2 this is just the photopeak efficiency of the gamma detector with a small correction for the anisotropy of the gamma-ray angular distribution, E. Auxiliary Experiments An alternative method of measuring the probability for Coulomb excitation is to detect the inelastically scattered particles by direct magnetic analysis of their energy. This method was 114 employed in the case of Cd to provide an independent cheek on the particle-gamma coincidence experiments. The results were ' found to corroborate those of the coincidence measurements, A description of the magnetic analysis experiment and its results is given in Appendix II, A. For the purposes of analysis it was necessary to know some of the E2 matrix elements to the higher or ''two phonon" states. In the cases of 11404 and 116 44 this information was available from the work of McGowan, et al., (1965). Only some of the matrix clements required for the tellurium isotopes had been measured ~ previously (Gangrskii and Lemberg, 1962); in particular, the value of B(E2, 47 2") was unavailable, A short experiment using the method of gamma-gamma coincidence was performed to obtain these values, This is described in Appendix IL B. 25 An attempt to extend the measurements to include bombardment by a 32, ion beam was made. The difficulties encountered and results obtained are detailed in Appendix I. C. The measurements and calculations performed with regard to the efficiency of the gamma-ray detector and the anisotropic angular distribution of the de-excitation gamma — rays are presented in Appendix IIL 26 IV. ANALYSIS AND RESULTS A. Corrections Applied to the Raw Experimental Data The experimental quantity obtained directly from the measurement is the ratio R, defined in (29), The quantity, Poxp? from which the quadrupole moment of the first ar state, Qo, is to be determined” » is given by do ) Poxp = (30) x. do. j=l 21 and is called the probability for Coulomb excitation of the first 2" state. A number of small corrections were made in order to — extract P exp from the raw experimental data. 1) Since the Coulomb excitation cross section is strongly energy dependent, the effective bombarding energy was taken to be the incident ion energy, E, minus half the target thickness, 1/2 AE; the cadmium targets were about 120, 800, and 1400 keV thick for Sue, 12, and 16, ions, respectively and the tellurium targets were about 40 and 500 keV thick for “He and 165 ions, respectively. A calculation of the effective bombarding energy correct to order (AE /B)" proved the above method to be sufficiently accurate. 7) One could just as easily analyze the quantity R, but analysis of the quantity P has the advantage of exhibiting more clearly the effect of Qo on the cross section for the first excited state. 27 2) The anisotropy of the gamma-ray angular distribution necessitated a correction of about 9% to the absolute yield in normalizing the Nal crystal efficiency to 4n geometry. This correction, however, was essentially the same for all bombarding conditions used in the experiments, For large scattering angles the gamma-ray angular distribution depends only weakly on the type and energy of the particle which causes excitation of the state. Also, the effect of the quadrupole moment on the gamma-ray angular distribution is negligible for large scattering angles. The coefficients for calculating the gamma-ray angular distribution are obtained from the computer program mentioned in Section IL, 3) The target nucleus possesses a recoil velocity after the collision with the projectile. This velocity, however, is reduced as the recoiling nucleus moves through, and loses energy in, the target. — if the target nucleus decays while still in motion, the effective solid angle of the gamma detector will be altered slightly. The maximum possible change in this efficiency occurs for 165 ion bombardment of a nucleus which then decays with its full recoil velocity, and this change amounts to only 0.3%. The application of this type of correction to the individual cases is discussed in Appendix I, 4) The contribution of cascade gamma. rays to the coincidence yield R must be taken into account, This contribution can be calcu- lated given the cross sections do, to the higher states and the cascade- to-crossover branching ratios, The finite width of the gamma-ray window may also permit acceptance of some or all of the gamma rays from the upper member of the cascade transition. The cross sections ds, for the higher states were calculated with the computer program 28 using the E2 matrix elements M,.. given in Tables land 2, These matrix elements were taken from the work of McGowan, et al., (1965), for the cadmium isotopes; the derivation of the matrix elements for the tellurium isotopes is given in Appendix IL B. orl was applied to the A correction of about 0. 5% for coincidence yield due to the de-excitation of the first excited state by the internal conversion process, This correction was, of course, the same for all bombarding particles, 5) The closer proximity to the single-channel-analyzer threshold of pulses from the inelastically scattered particles, and the pulse~height tail of the particle detector, resulted in slightly different efficiencies for detection of the elastically, and the in- elastically, scattered particles. A correction for this was deter- mined from the shape of the particle spectrum at each bombarding energy. This correction was made with great care since - at 4- 5% for 4 He bombardment - its magnitude was comparable with the deviations being measured, A coincidence-gated particle spectrum (see Figure 13) showed the same peak-to-tail ratio as the singles spectra indicating that the counts in the pulse-height tail of the Singles spectra above the single-channel-analyzer threshold were indeed substantially due to scattering from the relevant target isotope. An additional small correction was made to the particle count for particles scattered back into the annular counter from the tantalum beam stop, which was about two meters beyond the target. 6) The center-of-mass motion also introduccs a small correction to the coincidence yield because of the difference in solid angle subtended by the particle detector for the elastically, and the inelastically, scattered projectiles. This correction amounted to 29 ‘about 0.6% for 16, ion bombardment of 1i4ag and was corre- spondingly smaller for the other ions used, 7) The “Te target contained a total of 3% of other Te isotopes. The known gamma-ray energies and values of B(E2, 0° + 2*) of 14re, 18 re ang 18° 1956; Stelson and McGowan, 1958) enabled a correction (1. 5%) to be made to the coincidence yield, This correction, however, was the same for tHe and 165 bombardment, | Te (Temmer and Heydenburg, With the application of the above corrections to the measured ratios R, the experimental values, P... pAy E), of the probability of excitation of the first 2° state were obtained as a function of bom- barding particle and energy. These values are listed in Tables 3 and 4, B. Presentation of the Data A convenient way of presenting the experimental data to show the effect of Qe is to compare the measured values of P with those calculated from first order perturbation theory (28), This is done in Figures 14 through 17. The ratios P for bom- /Pres exp’ first order bardment by the various heavy ions are plotted versus &. The values of B(E2, 0° + 2°) used in calculating P, were taken from an analysis to be described later. irst order C,. Higher Order Effects The large deviations from unity in Figures 14-17 indicate that matrix elements other than (or | (E2)||2*) must also play a significant role in the 30 ' excitation process. These other matrix elements are those which, in the language of perturbation theory, lead to "higher order effecis'. It will be seen that the dominant higher order cffect arises from the existence of an appreciable static quadrupole moment in the first excited state of the nuclei studied here, The other higher order effects which can contribute to the - excitation of the first 2” state are virtual transitions via the higher- lying 2” levels, These, however, may be accounted for by direct calculation provided the pertinent matrix elements Mog are known. The magnitudes of these have either been measured or estimated and are given in Tables l and 2, The relative signs of these matrix elements are not known experimentally and this causes a considerable uncertainty in the deduction of Qo. in order to illustrate the above points and give a quantitative indication of the dependence of the cross sections on the projectile and its energy and on the magnitudes and signs of the matrix elements | M,,,, the results of computer calculations of the cross sections, do, ? for various sets of inputs are given in Table 5 8) The sign of Myo was taken to be negative in all cases. The last column of Table 5 shows in order the values of P derived from (i) a computer evaluation including all the levels shown in 114 Cd, (P ); Figure 11 for comp (ii) a similar computation including only the ground and first éxcited states, Prioo level’? (iii) first order perturbation theory, (28). 8) The computer calculations for Table 5 were done at Rutgers by J. de Boer. 31 A study of Table 5 shows the following: . a) Apart from the matrix element Myo connecting the ground state to the first excited state, the largest contribution to the cross section for the excitation of the 2” state arises from Moo; which is proportional to the static electric quadrupole moment Qo of the first 2" state (26). Changing the value of M,. from 0 to +1, 0 (e. 10724 em) decreases the cross section doy/ da by about 13% for 40 MeV O ions. The decrease is roughly proportional to Moo. b) The change in the excitation cross section do./ dQ arising from a change in the signs of M, 4 and My amounts to 3.8% and 1.3%, respectively, for 165 ions. The fact that the signs of M, 4 and M7 are unknown thus contributes a considerable uncertainty to the final analysis, The strong dependence of the cross section doo/ dQ on the signs of these two matrix elements arises from the possibility of excitation of the first 2” state via virtual transitions through one of the higher gr states. The interference of these virtual transitions with the direct transition from the ground state gives, in second order perturbation theory, contributions of the type pil2) ort _, 9+ to the excitation of the first 2” state which are proportional to the third power of the interaction constant Xpoee These contributions have the same dependence on the projectile and its energy as the coniri- bution p\l2) ot, It is to be noted that changing the signs of both M, 4 and M, A produces only a very small effect on the cross sections; this is because the interference term pul?) 91+ 497 is proportional to the product gts which involves the static quadrupole moment, Mio My4 Mog . The excitation of the first 2 state by virtual transitions through higher states with spins different from two, i.e., zero or four, must also be considered. However, this process can only 32 produce interference effects involving terms of the order of x", which are small and contain no ambiguity in sign, c) Inclusion of the cascade-to-crossover branching ratio c, (Stelson and McGowan, 1961) to calculate the cascade gamma-ray contribution, no, )) 7 1 do. (31) i=3 TC; 1 to the population of the first 2” state yields values of about 4% for 40 Mev 10 ions, 3% for 30 Mev 1c ions and 1% for 10 MeV 4H ions. This contribution is essentially independent of the signs of the matrix elements and absolute values of these are known to about + 20%. ad) The E2 matrix elements between states in the two- phonon region have a negligible effect upon the excitation of the first or state. This has been illustrated in Table 5 by changing the value of M45 from 0to1,0. In the final analysis one is therefore justified in setting Mg = 0 forr,s.> 2, e) The difference between Peomp and Prrst order 2rises only partly from the inclusion of the higher excited states in the calculation of P, omp* Computer calculations including only the ground and first cxcited states and with Moo = 0 yield values of P two level which ater from Prirst order as 7% for 40 Mev O bombardment. Deviations of this order of by, for example, as much magnitude are, however, expected because of the decreasing accuracy of a first order perturbation treatment as X14 2 becomes larger. 33 D. Least-Squares Fit to the Data The experimental values for Myo and Moo were deduced by comparing the measured excitation probabilities with those calcu- lated by the Winther-de Boer computer program, This comparison was done in the following manner, a) Preliminary values of Myo and Moo, denoted by My» and M55 , were obtained by using second order perturbation theory to analyze several of the data for 16, ion and tHe ion bombardment, This procedure is illustrated in Section II and Appendix IC. b) For a given projectile, Ay; and bombarding energy, E, the expression Pi hAys E) = a,(A,, E)(M, 5)" + a{A,, E)(M,.)* Moo (32) was taken to represent the dependence of the excitation probability on Myo and Moo. This is, of course, the prediction of second order perturbation theory. The coefficients ay and Ao, however, were not taken from second order perturbation theory, but were obtained from the computer program as follows. c) Two values of P nA E) were calculated with the computer program, first using Mi 9» M55 and M5 as input, and then using M5 + My», My + AMy5 and M..3 as input, where A << 1; the values of M.,: the matrix elements to the levels above the first excited state, were taken from Tables 1 and 2. Equation (32) was used together with the above values of PrylAy: E) to solve for ay and aos 34 The values obtained in this manner for ay and Ay depend on the relative signs of the M,..» and also depend (slightly) on the particular values of Mo and Myo (although this latter dependence vanishes in the limit of second order perturbation theory), However, for small changes in M,, and M,. about Mj» and Moyo, (32) reproduces to a high degree of accuracy the calculation of P.lAy, E) by the computer program. d) Steps b) and c) were repeated for all the bombarding — particles and energies used in the experiments, thus obtaining a set of equations (32), each linear in (M,,)° and (My 9)” Moo. The method of least squares was then used to obtain the value of M5 and Moo which, when inserted in (32), best reproduced the set of experi- mental excitation probabilities, Poxp 41 E). Because of the dependence of ay and Ao (and hence of the quadrupole moment) on the relative signs of the M.. a separate ’ least-squares fit was performed for each possible sion combination, Thus, in the case of 1404 four values of M,> and Moo have been deduced from the data. The fitted values of M values of Myo and M, analysis and used to calculate ay and Ao The resultant error in the quadrupole moment has been carefully estimated and was found to be © less than 5% in all cases, This error could be eliminated by an iteration of the fitting procedure or by using a more accurate expression for PinlAys E) than is given by (82). Such an expression is 12 and Moo differed slightly from the 9 which were obtained from the preliminary _ 2 2 Pin = ay + a, (M, ») + ao (M, 9) Myo . (32a) 35 ' However, in view of the larger errors (+ 30%) in Q, arising from other sources, it was felt that the small improvement in Qo afforded by an iteration of the analysis would not justify the cost of the necessary computing time. It may be noted here that the value of Myo used to calculate the a, and ao has itself a minimum uncertainty of + 8% which arises from the uncertainty in the NaI crystal efficiency. This introduces an uncertainty of about 6% in the quadrupole moment. E. Results The solid curves shown in Figures 14 - 17 represent the ' least-squares fits to the experimental data obtained by using the values of M.. s given in Tables 1 and 2 and with an ers except Moyo taken negative. Figures 15 and 16 for Cd and Cd both indicate a decline in the values of Pox of Pes est order at the higher | He energies, A possible explanation of this is the onset of interference from nuclear reactions as the top of the Coulomb barrier (~ 15 MeV) is approached, These higher energy points (dotted) have accordingly excluded from the analysis. This has also been done for the tie data in the tellurium nuclei, although the departure of the yield from Coulomb excitation theory appears to be more gradual than in the cadmium isotopes. The deduced values of Myo and Moo are given in Tables 6 and 7, Here and in Figures 14 - 17 the errors include only the counting statistics. The total errors for the various values of Myo and Moo deduced for each nucleus are given in parentheses in Tables 6 and 7, For M19 the predominant contribution to the total error arises from the uncertainty in the Nal crystal efficiency (+ 8%), 36 _ while for M,, the uncertainties in target thickness, M5 and the higher matrix elements each contribute to the total error, Tables 6 and 7 show that the value of Moo determined from the data may change by up to 30% depending upon the various choices of the signs of the M.,.. while the quality of the fit, indicated by the value of chi-squared, is similar in all cases. It is therefore not possible on the basis of the present experiments to determine these signs. This is due to the fact that the interference term responsible for this sign dependence is Similar in behavior to the interference term arising from a finite quadrupole moment, It can be seen from Table 6 that, despite the fact that p depends only weakly on the bombarding energy (see (24) and Figure 2), an analysis using the oxygen data alone is capable of yielding a value of Moo which has the correct sign and order of magnitude. It should also be noted that the value of Mio derived from the analysis is rather insensitive to the signs of the M.. 5° In principle the B(Z2) and quadrupole moment can be determined from any two experimental data, either of different energy or of a different bombarding particle, though the latter constitutes the more sensitive measurement, The fact that a good theoretical fit is obtained to the cadmium data points of three different bombarding particles, covering a wide range of energies, attests to the validity of the theory in describing the excitation process, Only two different ions were used in the measurements on the tellurium nuclei since the quadrupole effect itself was smaller and the validity of the method had already been established in the | cadmium measurements, 37 To Single out the effect of Qo, the data and fits of Figures 14 - 17 are replotted in Figures 18 - 21, normalizing now to Pp ¢ omp ‘22 = 0), Poomp 22 = 0) is a computer calculation of P using the fitted value of My» and all the other matrix elements in Tables 1 and 2 but with Moo = 0, Thus, if the quadrupole moments had been zero, all the data points in these figures would be expected to yield, in the least-squares fit, a common line of ordinate unity. — The slope of the 16, curves in the figures in which Poxp is normalized to Por st order is due partly to the decreasing accuracy of first order perturbation theory as the excitation probability increases, From the above discussion and the similarities of Figures 14 - 17 and Figures 18 - 21, it is clear that the major part of the deviations from first order perturbation theory may be attributed to non-zero quadrupole moments of the first 2° states in these nuclei. 38 V. DISCUSSION OF THE ANALYSIS The preceding analysis rests on certain assumptions concerning the excitation process; the validity of these will now be discussed. 1) The treatment of the Coulomb excitation process in the semiclassical approximation assumes that the projectile follows a classical hyperbolic orbit, an assumption which is valid for large values of n (see Section 0). At the energies used in the present experiments the values of n for the bombardment of 11409 with tHe, 12, and 165 ions are approximately 10, 30 and 40, respectively. The use of symmetrized expressions for the distance of closest approach and adiabaticity parameter further improves the accuracy of the semiclassical approximation such that, for the experiments reported here, the difference between a semiclassical and a quantum mechanical treatment is expected to be negligible (Alder, et al., 1956). 2) Only electric quadrupole interactions between projectile and target have been considered in the analysis. . Because of the selection rules applicable to the present case magnetic interactions can only contribute to higher order effects and not to the direct excitation of the 2” state. Such contri- butions are, however, expected to be 100 to 1000 times smaller than the E2 effects since magnetic excitations are intrinsically much weaker for particles scattered at large angles (the cross section is zero for 180° scattering) and for particles with non-relativistic velocities. 39 The effects of electric dipole interactions have been discussed by Eichler (196 4), He considered the term in the perturbation expansion arising from the interference between virtual double El excitation via the giant dipole resonance and direct E2 excitation from the ground state. In the type of experi- ment reported here, the expected dependence of the double-E1 effect on the various bombarding conditions employed is sufficiently similar to that of the static-quadrupole-moment effect that an experimental distinction is impracticable. However, according to more recent calculations by MacDonald (1964), Douglas (1966), and Winther (1966) based on the collective model, the magnitude of the El effect is expected to be 5 - 10 times smaller than observed in these experiments. Accordingly, the effects of virtual double E1 excitation have been neglected in analysis and also in the estimation of the error in Moo. The effecis of E3 and higher multipoles have also been neglected. Although a weak E3 excitation of a 3. state at about 2 MeV has been observed by McGowan, et al., (1965), the contribution of this process to the excitation of the 2” state is considerably smaller than the equivalent contribution from the second ar state, 3) Only the lowest known excited states (6 for **4cq, 4 126 128 i Te and ~~ Te) have been included in the for oq and 3 for computer analysis although only about 10% of the energy weighted E2 sum-rule is accounted for by the transitions to these states. This limitation seems justified on the grounds that other low-lying states which could contribute significantly to the excitation of the first 2” state should also have been observed previously through their gamma decay. Furthermore, if the remaining 90% of the E2 40 strength is accounted for by a single "giant quadrupole" state at about 10 MeV, and if a matrix element of equal magnitude is assumed to connect this state with the first 2° state, then the resulting interference term in the probability of excitation of the first 2” state in li4eg is found to be negligible. 4) The possible effects of nuclear reactions have been ignored. While it is extremely unlikely that such effects are important for the 126, and 165 ion bombarding energies used here, it is known (Hansen and Nathan, 1963, and McGowan, et al., 1965) that nuclear reactions may become appreciable for “He bombardment as the energy approaches the top of the Coulomb barrier (about 15 MeV for tHe on 11404), It is therefore especially important in experiments of this type to establish the energy range within which one may have confidence in a purely electromagnetic description of the interaction, This is the reason why 4H6 data were taken over a wider energy range than was ultimately analyzed in the present experiments, An examination of the analyzed 4H data points in Figures 18 - 21 suggests that a small but finite slope is present in the data for each nucleus. The significance of this slope, if real, is not understood. It has been assumed that the tHe energies used in the experiments and included in the analysis are far enough below the top of the Coulomb barrier to make interference from nuclear reactions extremely unlikely. The validity of this assumption appears to be proven by the simultaneous fit to the data involving the three different bombarding ions, It is noted, however, that if a monotonic decrease in the 4He yield with increasing energy due to nuclear reactions is actually present in the analyzed data, that 41 this would lead to an underesiimation rather than an overestimation of the quadrupole moment, It would be interesting to extend the *He data to higher energies up to and beyond the top of the Coulomb barrier to examine this assumption further, ' 42 VL DISCUSSION OF THE RESULTS Before discussing the results of the experiment, a few | comments on the collective model are necessary. The collective model of the nucleus in its simplest form classifies the collective or correlated motion of the individual nucleons into two basic types - rotational and vibrational motion, Rotational motion is characterized by the existence of a permanent equilibrium deformation; the nucleus may rotate about an axis perpendicular to its (assumed) axis of symmetry. Vibrational motion corresponds to an oscillation of the nuclear surface much the same as can be experienced by a liquid drop. While it appears possible that a nucleus may vibrate about an equilibrium deformation, thus permitting a "mixing" of the characteristic features of rotational and vibrational motion, the energies associated with the two types of motion are usually sufficiently different that nuclei may be empirically classified as either rotational or vibrational according to the structure of their lowest-lying levels. Rotational-type nuclei are found in the mass regions”) A= 150 to 190 and above A= 220. The energy level structure of 10) in these regions is characterized by a J(J + 1) energy Spacing and a sequence of J-values 0", ar qr 6", etc. , even-even nuclei 9) Examples of collective motion are also known in other regions of the periodic table, including probably the 1p-shell nuclei, 10) Only even-even nuclei will be considered in the discussion for reasons of simplicity and because the nuclei investigated in these experiments are all of the even-even type. 43 with the energy of the first excited state usually in the 100 to 300 keV range. . Vibrational nuclei, on the other hand, exhibit a quite different level sequence and spacing. A triad of excited states with J" = O°, 2", 4* is usually found at about twice the excitation © energy of the first excited state with J" = 2", The excitation energy of the latter is typically 400 to 700 keV. Vibrational nuclei are found in the mass regions A = 60 to 150 and A = 190 to 220, It is apparent from an examination of their energy level spectra that the cadmium and tellurium isotopes investigated here are typical vibrational-type nuclei, The simplest vibrational model considers pure harmonic quadrupole vibrations of the nuclear surface (Bohr and Mottelson, 1953). The equation of the surface may be expressed as ; | R(8, 9) = Rp (1 ‘2, ty 4 Zp (9) where the 02 are functions of time and are assumed small. By expressing the Hamiltonian of the system in terms of the generalized coordinates, Oo Ww and their associated generalized momenta, it may be seen that the surface oscillations are equivalent to a system of harmonic oscillators. The excitation quanta are Bose-Einstein particles or "phonons" of spin 2 and even parity. The first excited state is a one phonon state while the degenerate (0", a" 4") triad is obtained from the combination of two phonons. In the simple harmonic model (SHM) the matrix element of the E2 operator between states containing the same number of pure phonons vanishes identically, Thus, in this model, Q, = 0. 44 The fact that the experimental values of Qo determined in the present work, and also recently by others?! , are not zero is perhaps not too surprising, however, when one notices that the SHM provides only an approximate description of the actual nuclear motion, An examination of the known properties of most vibrational nuclei immediately reveals deviations from the SHM. (For the remainder oi the discussion 114 0G is taken as the case in point), The degeneracy of the two-phonon triad is removed and all of the levels are seen to lie at more than twice the energy of the one-~ phonon state, Furthermore, an examination of the matrix elements in Table 1 shows that the selection rule forbidding E2 transitions between states differing by other than one phonon is violated by the small but finite value of B(E2, 2 +0"). There are also other similar examples of deviations from the pure-harmonic phonon model, It may be hoped that a more refined model which can account for some of these deviations will also predict the measured value of Qo. Within the framework of the SHM, deviations from harmonic structure are accompanied by the admixing of states of different numbers of phonons, In other words, the 2° .0° crossover 11) Measurements similar to those reported here have been done on 11404 also at Oak Ridge National Laboratory by Stelson, etal., (1965 ) and at A, W.R,E,, Aldermaston by Simpson, etal., (tobe published). Their values for Qo, -0.6 + 0.2 barn (Oak Ridge) and -0,49 + 0,23 barn (Aldermaston), agree within the experimental errors with the value reported in this thesis, -0.7+ 0,2 barn, A part of the experiments described here has been reported in the literature (de Boer, Stokstad, Symons and Winther, 1964, and 1965). 45 - transition is accounted for by the presence in the two-phonon state of a small admixture of the one-phonon state. Similarly the deviation of the transition moment ratio _ B(E2, 2'* + 2") B(E2, 2° + 0°) Ry from the SHM prediction of 2 is an indication of phonon mixing. Tamura and Ugadawa (1965) have taken the experimental value R, = 1.24 0.2 (McGowan, et al. , 1965) and used it to calculate the amount of two-phonon character present in the one-phonon state on the basis of a simple model. In their model the wave functions of the first and second excited 2” states are written respectively as (2) = a,|1) +a,|2) y(2') = -a,|1) + a,|2) with a.” 264 14 7» . The one-phonon amplitude ay and hence the admixture amplitude a) are fixed by the relation a(2a,* - 1)? ST ay 46 yielding at = 0,86 and as = 0,14. The magnitude of Q, is given by 12 1 2 Qo Gay aja, 4Rq 8 where §, a parameter related to the mean square amplitude of the surface vibration, is derived from the B(E2, 0° - 2") and is equal to 0.2. Evaluation of the above yields 1Q, | = 0,58 barn, The sign of Q5 is not predicted in this simple model but depends on the nature of the force producing the mixing. It would appear that the problem is solved by this simple model, This, however, is not the case. The quadrupole moment has been explained at the expense of the forbidden crossover transition. The model predicts B(E2, 2" = 0°) _ 0. 14 B(E2, 27 +0 Ro whereas the experimental value is p{ex) = 0,015 + 0, 005. The phonon admixture of 14% which explains R, and Q, produces in this model a crossover transition rate almost an order of magnitude too large. It is also possible that such a large phonon admixture violates the original assumptions on which the model is 47 12) Rather than considering at this point the other vibrational based, One cannot therefore regard this model as satisfactory model calculations, all of which incorporate individual particle structure to some extent, it is advantageous to look to another collective model, the rotational model, as it is known to predict large quadrupole moments. In the simple rotational model, all transition and static E2 moments are related by Clebsch-Gordan coefficients, In particular, [Q,| = 0.91] (0 |] m(E2)|/2 > | The value of B(E2, ov. 2") is known and the above relation gives |Q,| = 0.7 barn which agrees with the experimental value of -0,7+4 0, 2, The rotational model predicts Q»5 negative if the intrinsic deformation is prolate. Here one can see the fundamental theoretical difficulty posed by the measured value of Qo; 11404 is a nucleus displaying what appears to be a vibrational character in all respects except one, the value of Q,, which is rotational in character, A rotational model capable of accounting for some ap- parently vibrational characteristics is the asymmetric rotator model of Davydov and Fillipov (1958), They consider the rotations of a permanently deformed, ellipsoidal nucleus with three unequal principal axes. The deviation from axial symmetry is expressed in terms of the parameter y which is empirically fixed by the 12) Neither do Tamura and Ugadawa. They wish only to show by means of this model that a vibrational description can be found which accounts for Q9. Whether this model actually accomplishes this is open to ‘question. 48 114, d, ratio of the energies of the first and second 2” states. For y = 26.8 degrees, One other parameter is needed to determine Q. ; this is Qo, the so-called "intrinsic" quadrupole moment, which is related to the B(E2,0° - 2"), The Davydov-Fillipov model predicts | Q,| = 0.3 barn . Q. is negative if one assumes a prolate equilibrium deformation, i,e., assumes @) positive. Although this value of Q5 is still too small, the asymmetric rotator model gives a more reasonable description of 11404 than the simple rotational model which has no second-excited 2* state. A difficulty with the Davydov-Fillipov model, however, is the lack of a 0” state to complete the two-phonon triad, which is frequently seen experimentally. To summarize the above discussion, one may note that none of the phenomonological collective models considered thus far - the pure harmonic vibrator, phonon-mixing, simple rotational model and the asymmetric rotator models - has been able to explain all of the major charactcristics of 11404, A suggestion of the next direction which one might take in searching for an explanation of this dilemma is provided by the presence of the additional 0” and 2” states in the two-phonon region - of 1i4ag (see Figure 11). These are presumably intrinsic states arising from the excitation of individual nucleons and should serve as a reminder that individual particle aspects can influence the collective properties of the nucleus. The simplest model based on individual particle motion is the ordinary shell modcl (Mayer and Jensen, 1955). For the cadmium nuclei (Z = 48) one may take as the relevant shell model 49 contigueation for the first excited state two proton holes ina (So jo 2 configuration coupled to angular momentum 2, Evaluation of Q5 (Tamura and Ugadawa, 1965) gives” Q, =-0.10 bar. The sign is predicted correctly!®) but the magnitude is too small, This disagreement should not be taken too seriously, however, since one should not expect such a simple shell model description to work where collective motion is so clearly important. A more realistic approach can be taken by coupling individual particle motion with collective surface oscillations (Bohr, 1953). In such a model two nucleons in the shell model configuration (nt 3)? with principal, orbital and total angular quantum numbers n, t and J are coupled to a phonon of spin R. The phonon and the shell model configuration states are then coupled to total angular momentum I=3+R for the nucleus. Detailed calculations by Macdonald (1963) yield , However, in this model the 4* state occurs below the second 2° state and the 0° state lies too high. it is also possible to treat the collective motion on a microscopic scale. The nucleons outside the last closed shell are 13) This model, however, would predict the wrong sign for the tellurium isotopes. 50 ~ coupled by a pairing force, and the coordinates of the paired particles are suitably transformed to those of quasi-particles in a manner directly analogous to that used in the theory of superconductors, A long-range quadrupole-type force is then used to couple the quasi-particles and produce a collective motion, The resulting equations are then "linearized" i.e. , treated in various degrees of the random phase approximation (RPA), RPA calculations by Tamura and Ugadawa (1965) yield quadrupole moments on the order of Q =-0,08 barn . The reason for this small value of Qo is that the ordinary RPA: treatment neglects precisely those diagonal terms which are responsible for producing a quadrupole moment. Noting this, Tamura and Ugadawa (1966) have made RPA calculations in which the otherwise neglected terms are treated as a perturbation. Their results show a more reasonable value of Q» = -0,44 barn .. This model, as have all others, experiences certain difficulties, It predicts 1Q,, 11404) ) 12, 11604) the opposite of which is observed experimentally, Also, it predicts a positive value of Q, for the tellurium isotopes, contradicting the experimental values. 51 Other authors (Dang, et al,, 1966) have also made attempts to calculate the large quadrupole moments observed in 1144, These calculations differ in some respects from those proposed by Tamura and Ugadawa but have in common the facts that i) they are only simplified, exploratory calculations and are not to be taken as final theories and ii) they both indicate that phonon mixing and particle mixing may provide the mechanism for producing a large quadrupole moment. The present state of theoretical interpretation of the large quadrupole moments observed in 11404 and 11644 is thus seen to be far from satisfactory. Whether the final solution will be found in terms of a vibrational model description or within the context of | some different, as yet unproposed, type of model must remain for the moment a matter of conjecture. A summary of the present experimental resulis, together with some of the other known properties of the cadmium and tellurium isotopes is presented in Figure 22. This figure shows the ratios B(E2, 2" - 2") *B(E2, 2° = 0°) ? B(E2, 47 + 2°) B(E2, 2* 0°) * 52 E Et , and Mga _¢2* |] moe2)j2*) IMyoT | (atl m (ea)l|o*y} It is apparent from Figure 22 that the variation in magnitude of the ratio My./ [MI for the four nuclei studied here follows a pattern suggested by the other properties of the cadmium and tellurium isotopes, In particular, as the transition moment ratios and energy level ratios approach the value 2 predicted by the SHM or "phonon" model, the magnitude of M,,/|M,.| decreases toward the SHM prediction of zero, Tables 3 and 4 show also that the magnitudes of Q, for the tellurium isotopes are smaller than those of the cadmium isotopes; the neutron numbers of the latter are midway between the major closed shells at N = 50 and N= 82, Finally, it is noted that the sign of Qo does not change in crossing the closed proton shell at Z = 50. In summary, the quadrupole moments of the first excited states of 4cq ana 11664 measured in these experiments, are surprisingly large. How large they are is emphasized by the fact that 11404, a nucleus which otherwise shows characteristics of a vibrational nature, possesses a static electric quadrupole moment in its first excited state which is as large as would be obtained with a rotational model description. Theoretical models, introduced prior to the measurement of Q, to account for the other deviations 53 ' from vibrational-type structure, predicted a value of Q, which is too small, Models introduced subsequently, and designed to predict a large Qo, oupersence ditticulty in reproducing the other known properties of Cd. The systematic variation of Qo for the four nuclei studied follows a pattern which appears to be consistent with the trend of the other observed deviations from pure harmonic structure. No satisfactory theory exists to describe all these deviations from pure harmonic behavior; the failure of the existing theories to explain the large values of Qo is especially striking, 54 APPENDIX L SYNOPSIS OF THE MULTIPLE COULOMB EXCITATION COMPUTER PROGRAM The computer program for multiple Coulomb excitation written by Winther and de Boer (1965) provides the experimentalist with an essential and convenient tool for the analysis of data obtained from this type of measurement, Although the input requirements and use of the program in the analysis for this particular experiment have already been mentioned in Section II, it is worthwhile to examine in brief some of the internal aspects of the program. | The program uses the semiclassical approximation (7) together with the E2 multipole interaction Hamiltonian (8) to solve for the amplitudes, a, ( ), of the free nucleus eigenfunctions. The solution is obtained by direct numerical integration of the coupled differential equations (7). The variables ¢ and y(t) in (8) are replaced by t == (ec sinhw + w) <lp x(t) = a(e coshw + 1) where e = (sing) such that (7) becomes 55 da._.(w) | . 2 wo Fi) Qy (Er Sng) KCL, x ag Cw) (33) H,s where cow a,e M5 TS vat It is noted that Crs is proportional to the interaction constant , defined in (12). “The probability for the Coulomb excitation of a state ‘N from an unpolarized ground state (I,) is given by PO) = aa d | \? (34) M,,M and the Coulomb excitation cross section for an unsymmetrized calculation would then be obtained from (2). Ih determining a symmetrization procedure for a multiple excitation calculation, the conditions of time reversal invariance and unitarity were satisfied in addition to the requirement that the symmetrization reproduce the limit of a quantum mechanical calcu- lation, A symmetrized form of Bag? gSymm , analogous to (27) was rs used in the factor Q ules bg? w) in (33). 3 56 1/2 Z.ZiA . symm _—1° 2°71 empty l/2ipn giy7l/2 ors ~ 6, 235 {le EL [E E,! } (35) where E.. = (14+ A,/A,) (E.. - Ei) (36) and E, E.. and E. are the energies in MeV of the projectile, state r and the initial state, respectively. The procedure for symmetrizing ¢... was to replace v everywhere with VV where rs v= (37) with M the mass of the projectile. Half the distance of closest approach, a, is replaced by ang where Zs Zoe" an = -— (38) rs mviv ors - This symmetrization procedure is exactly equivalent to an internally consistent shift of the energy of each level and a renormalization of the E2 matrix elements M.g° The symmetrized calculation is thus equivalent to an unsymmetrized calculation performed for a slightly different nucleus, Unitarity is therefore not violated by the intro- duction of symmetrized parameters and r P(N) = 1. . 57 The requirement of time reversal invariance is given by the relation (21, + Ive do(N + M) = (21, + 1)ve, do(M + N) (39) for the cross sections for the excitation processes N -M and M-N, An examination of the differential equations (33), in which symm rs shows that the equations are invariant under the conditions of the symmetrized parameters & and ow are employed, reversed motion; reversed motion is the situation in which the nucleus in state f{ is bombarded by a projectile of energy E - Ey . This invariance results in the relation yy (Mi) = ayy OM) (40) {f£ — 1i from which follows (21, + 1)P(N-™M) = (210+ 1)P(M-N). (41) Using (41), it is easily shown that 2 Vv do(N + M) = ine Pon, cSymm, symm) “NM gi74(8) (49) satisfies (39). 58 P(2) as defined in (42) is seen to be slightly different from Poxp = do,/ , do, as defined in (30). The analysis has been performed in terms of the quantities actually measured, namely the ratios of cross sections as given by P exp? rather than in terms of the excitation probabilities defined by (42). It is noted that the symmetrized expressions I C. 15 in Alder, et al., (1956), (28), and (42) are all equivalent in the limit of first order perturbation theory. The integration of (33) starts from the initial condition that all a, (w = -U,) = 0 except for the ground state amplitude. The range, Uy and initial step width of the integration, 24w, are determined by the required absolute accuracy which is stated in the input. A Runge-Kutta-Gill procedure is first used to generate the amplitudes and their derivatives for four consecutive steps. With these values as starting points, the faster Adams- Moulton predictor-corrector integration routine is then used to complete the calculation. A constant check on the accuracy of the integration is maintained and an adjustment in step width executed whenever necessary. . The differential cross sections for each level may be obtained either in the laboratory or center-of-mass systems. The program also calculates the coefficients for the gamma-ray angular distribution in various coordinate systems. The formulae given are sufficiently general to include the fact that, in multiple . Coulomb excitation, the various levels may be populated not only by direct excitation from the ground state, but also by electron conversion and gamma-ray transitions from higher levels. 59 APPENDIX I. AUXILIARY MEASUREMENTS A. Magnetic Analysis Measurement It is also possible to detect Coulomb excitation by direct magnetic analysis of the energy of the scattered projectile. This mcthod is particularly suitable when protons or alpha particles are employed as projectiles, since the energy losses in the target remain small. This type of measurement becomes more difficult when heavier ions are used; large dE/dx energy losses for heavy ions require an extremely thin target in order to be able to resolve the elastically and inelastically scattered particles. A measurement by magnetic analysis was made, nevertheless, in order to have an independent check on the particle- gamma coincidence method, The magnetic analysis method offers several advantages over the particle- gamma coincidence method. (i) It is not necessary to determine the efficiency of a Nal crystal to extract the excitation probability. (ii) The population of the first 2” state by cascade gamma - rays is no longer a factor to be considered in the analysis, (iii) The measurement does not require electronic coincidence circuitry with its accompanying dead- time losses and random-coincidences. The principle difficulty encountered with this technique is the low counting rate, 60 Data were obtained by this method for the scattering of He and 16, ions from 11404 at a laboratory angle of 150 degrees, The 4H6 and 165, energies used were 9, 76 and 40.7 MeV respectively, both corresponding to § = 0,30. Adequate resolution of the inelastic and elastic particle groups was obtained by using thin targets (200 up gm/ om” for *He and 20 pgm/ em” for 165 bombardment) and a small spectrometer acceptance angle (about 4 2° total) to minimize the kinematical variation of particle energy 11464 at 150°), The scattered particles were detected initially by a linear array of sixteen solid-state detectors placed in the focal plane of a 61-cm with angle (about 60 keV per degree for 185 on double-focusing magnetic spectrometer. The total energy region spanned by this detector array was quite adequate to cover the region of interest but, because of the wide gaps between the detectors, it was still necessary to scan across the spectrum in several overlapping steps. A typical momentum spectrum obtained with 4 He projectiles is shown in Figure 23. For 165 bombardment, however, the counting rates were so low that it turned out to be impossible to obtain adequate statistical precision with the counter array. After taking an exploratory spectrum with poor statistics (see Figure 24), the sixteen counter array was replaced by a Single large detector, collimated by a 1.27 cm slit. This detector spanned an energy region of approximately 200 keV, sufficient to cover either the complete inelastic peak or the complete elastic peak for one choice of the spectrometer magnetic field. By varying the magnetic field of the spectrometer, the detector was set on each peak separately; the number of particles scattered by the target was monitored by a solid state detector placed in the scattering chamber at an angle of 150° on the opposite side of the beam axis, The back- 61 ground under the inelastic peak was estimated by counting on either side of this peak, This subtraction was about 2% of the inelastic yield. In calculating the ratio of the cross sections for the 16, case, it was necessary to allow for the small difference in the | charge state populations for the elastically and inelastically scattered particles, This difference was measured by determining the yield of elastically scattered 165 ions in the selected 6" charge state as a function of the energy of the incoming beam. The difference between the 6” charge state populations at the energies of the elastic and inelastic groups was measured to be 1. 5%. The results of the spectrometer measurements are given in the following table. The errors are almost entirely due to the limited statistical precision of the number of counts in the inelastic peak, Projectile EMev 4 Pexp x 1000 16 O 40. 70 0. 301 T5+4 4H 9.76 0. 301 6.7+ 0.2 The analysis of the above experimental quantities was done in a manner similar to that described in Section IV and yielded the values M,, =-0.709+ 0,012 e- 10°?4cm? 12 24,2 My, =+1.04+ 0.3 e+» 10 cm 22 These values compare favorably with those given in Table 6 for the particle- gamma coincidence method, 62 B. Gamma-gamma Coincidence Measurement The analysis of the data required a knowledge of the matrix elements M..s connecting the levels populated in the experiment. Gamma-gamma coincidence measurements performed by Stelson (1961) and McGowan et al. , (1965) provided this information for the. cadmium nuclei, Only part of the required information was available r 126.06 and 128.16, The values of B(E2, of 5 2°) and B(E2, of 2'*) for 126.4 had been measured by Stelson (1958) and Gangrskii, et al., (1962), respectively. These values together with the known cascade-to- fo crossover branching ratio (Perlman and Welker, 1954) determine the B(E2, gr a*) since the 2't + 2° transition is known to be 98% E2 (Wiedling, et al., 1960)!4), | In order to obtain a value for B(E2, 4° + 2°), the following experiment was performed, A Te target prepared on a clean gold backing was bombarded with 42 MeV oxygen ions. Two gamma- ray detectors - one, a 10, 2-cm-long by 12. 7-cm-diameter Na(T1) crystal and the other, a 20-cc lithium-drifted germanium detector - were mounted on either side of, and close to, the target, The germanium detector spectrum was gated by gamma-gamma coincidences and the pulses from the Nal detector were required to fall within a window set around the 667 keV (2° - 0°) photopeak. The Li-Ge detector spectrum is shown in Figure 25; it consists 14) The SHM predicts the 2'" - 2” transition to be pure E2, Gamma~-gamma angular correlation measurements by Lindquist and Marklund (1957) on a variety of nuclei show that the a'+ 29° transition is generally greater than 90% E2. However, one of the exceptions was Te with a 50+ 10% Mi multi- polarity. The decay of 2276 was measured to be 92% E2. 63 almost entirely of a background of real coincidences whose source was most likely the 126 aor p a) Na reaction from the carbon accumulated on the target during bombardment. The peak at 667 keV indicated by the arrow in Figure 25 is due to the 2° ~ 0° gamma rays from the 4° - 2* +07 and the 2'* - 2* + 0° cascade transitions. Their presence in the spectrum arises from the fact that the window’ in the NaI detector spectrum also accepted the upper member of the 4" +2740" cascade and a fraction of the upper member of the art 4 o* ot cascade, The estimated random coincidence contri- bution to this peak was negligible. | The horizontal bar indicates the expected position of the at assumed to be due to the 4" - 2° transition, and if the 1,360 MeV state in 1*87¢ is indeed 4°, one obtains !®) a value of B(E2, 2° ~ 4°) = 0,234 0,10 e2- 10°48 om*, This means a value of B(E2, 4° = 2") -/B(E2, 2° = 0°) of 1.2 which is reasonable for these vibrational-type. nuclei (see Figure 22), This value has been used in the analysis. A similar experiment was not attempted for 12876 since +9 gamma ray. If the peak which appears under this bar is the energies of the upper and lower members of the cascade are so close as to make their resolution difficult even with a Li-Ge detector, To estimate the higher matrix elements for 128.76. the values of the ratios B(E2, 4° = 2*)/B(E2, 2* +0") and B(E2, 2" = 2*)/B(m2, 2°07) were taken to be the same in 787 ¢ as in 12°Te, The B(E2, 0 + 2") used in these ratios was taken from the work of Stelson (1958) and the value for B(E, ar 0*) was assumed to be the same as in 126 6, The resultant values of the M... for 12606 and 128.76 are given in 15) Since the face of the Nal crystal was only 1.2 cm from the target, it was assumed that the effect of the angular correlation between the 4+ = 2* and 2+ + 0* gamma rays could be neglected. 64 Table 2, It should be mentioned that these values are not overly critical for the analysis, especially since their relative signs are unknown, This latter uncertainty effectively puts an error of + 100% on these matrix elements, The error in Qo from this source is much larger than the corresponding error introduced by the cascade gamma-ray correction due to the uncertainty in the magnitude of M.. 5° C. Sulphur Beam Measurement In Section ID, 2 it was shown that the magnitude of the effect of Qo on the excitation probability is roughly proportional to the mass of the bombarding particle. It was decided therefore to attempt an extension of the measurements to include a still heavier bombarding ion, 325 Sulphur was a natural choice because it had previously been accelerated with the tandem and because it is significantly heavier than oxygen. Several difficulties were encountered in the production of the sulphur beam which seriously limited the utility of this method for measuring Qo Some of these difficulties, particularly those connected with the production of a negative ion beam, have already been described in detail by Miller (1966). Briefly, they were: {i) the introduction of H,S into the negative ion source had a detrimental effect on the ceramic insulators Separating the various acceleration electrodes in the ion source, After typically twenty-four to thirty-six hours of operating time, conductive tracks developed across the surfaces of the insulators; 65 (ii) although no oxygen was deliberately introduced into the duoplasmatron source a negative oxygen beam of 80 keV energy was present with the same magnetic rigidity as the 40 keV Sbeam. This had the effects of loading the tandem with an unwanted beam and making the selection of, and regulation on, the desired sulphur beam more difficult. Post-terminal stripping produced a number of beams of intensity comparable to the desired beam at about the same terminal voltage; (iii) the low net intensity of the S beam from the negative ion source effectively prohibited the use of the 8” sulphur charge state, limiting the maximum energy to about 45 MeV. As this is less than one half of the Coulomb barrier energy, the excitation probability was rather low. This together with a low beam intensity (~ 5 nanoampere) produced a very low counting rate. It was nevertheless possible to obtain a measurement of the excitation probability of the first excited state in 11604 under bombardment by 32, ions. The negative beam was produced by insertion of argon into the duoplasmatron source and H,S into the exchange canal. Positive identification of the high-energy analyzed 32, beam was obtained by scattering the beam from thin gold and nickel foils and observing the ratio of the scattered particle energies, With the tandem regulating on the 7 charge state, a beam of four nanoamperes at 44 MeV was available at the target. In a net beam _ time of about five hours, 1280 coincidences were obtained, The methods described in Section II D, 2 suffice for an analysis of this datum. The experimental excitation probability 66 obtained was 0, 38 + 0.04%. The B(E2, or 2") obtained from a comparison of this value with first order perturbation theory was 0.55 + 0,05 e”. 107% om*. To obtain a value for Q, one compares the ratios of the experimental excitation probabilities of, for example, tHe ions and 32, ions with the theoretical ratios by means of the following equation 32 Poxp ¢ 9) _ pit, 1) 82g) (1. 5 22g) j Paxp Cre) pM) Aue) (1+ 9 (He) (sce (19), (22) and (24) ), Using the 4Ht6 datum taken at a bombarding energy of 9,02 MeV, one obtains 0.572 + 0, 044 = 0,616 (1 - 0,123 Myo) Moo = +0.59 + 0.59 and Qo = -0.45+ 0.45 barn. . The principle uncertainty in the value of Q, thus derived is due to the uncertainty in the measured target thickness. The ~°S ions lost 4, 1 MeV in passing through the particular target used. Clearly, thinner targets would be required for measurements using 32g ions, with even lower counting rates being the result, 67 APPENDIX GAMMA-RAY SPECTROMETRY It was necessary to know the efficiency of the NaI gamma detector in order to deduce the probability for Coulomb excitation, This efficiency was determined in the following way. Point sources . of gamma rays of various energies were prepared and placed ai a distance of 10 cm from the face of the 7,62-cm-long by 7, 62-cm- diameter NaI crystal. The total efficiency and peak-to-total ratios for the detector and source in this geometry were taken from the Nuclear Data Tables compiled by Marion (1960) and were used to determine the strength of the point sources. The calibrated sources were then placed at the intersection of the beam and target and used to measure the efficiency of the gamma-ray detector in the actual experimental configuration, The major uncertainty in the resultant efficiency is contained in the values of the total efficiency and peak- to-total ratios initially used to calibrate the sources. The uncertainty in the experimental efficiency was estimated to be + 8%. The anisotropic angular distribution of the 2” + 0" de- excitation gamma rays requires a correction to the isotropic-source efficiency measurcd above. The gamma-ray angular distribution is a function of 6 only, where 6 is the angle between the gamma-ray direction and the beam axis, because the particle detector has azimuthal symmetry. The probability, dW, that the nucleus in its first excited state will emit a gamma ray into a solid angle do at an angle @ is given by aw= 1+ ao Po (cosé) + ay Py (cose) . (43) 68 Po and P 4 are the Legendre polynomials. Typical values of Ao and a4 obtained from the computer program were 0,7 and -1.6, respectively. The finite size of the NaI crystal tends to attenuate the effects of the anisotropic distribution. The above expression for dW must be folded with the efficiency of the Nal crystal (as a function of 6) and integrated over the volume of the crystal. This procedure has been described by Pearson (1963). The value of the correction factor K which multiplies the isotropic efficiency is then given by Jo J4 K= l+a,s— P,(cos6,) + ay = P,(cos8,) (44) 2 Jo Jo where 84 is the angle of the axis of symmetry of the NaI crystal with respect to the beam axis. The factors J o/ Jy and J a/ Jo are referred to as Rose's smoothing factors (Rose, 1953). Since they are dependent upon the source ~ detector geometry, they were measured experimentally using a narrowly collimated source of gamma rays. The results agreed with those obtained by inter- polation of results found in the literature (Herskind and Yoshizawa, 1964). The correction to the isotropic efficiency due to the aniso-~ tropic angular distribution amounted to about 9% but was essentially the same for all bombarding particles and energies, The de- excitation gamma ray is emitted from the recoiling nucleus after a time ~ 10724 sec, which is long compared to the a1 sec.). The gamma ray will be emitted collision time (a/v ~ 10° at angle @ in a coordinate system moving with velocity v = Bc relative to the laboratory system, provided the velocity of the 69 recoiling nucleus is not appreciably attenuated by the target thickness. The gamma ray will appear to have an angle 6 in the laboratory system. For projectiles scattered at large angles, the component of the recoil velocity perpendicular to the beam axis may be neglected. A relativistic transformation of velocities then yields a= tan} (Sima. Ji-s). (45) cosé - B The net result of the change in angle is to introduce small P,, © Ps; and P. terms into the laboratory angular distribution, To first order in p, (44) becomes J J Aa, J J _ 2 4 ; 2, 1 6 2 3 0 0 0 0 J 5 7 +3247 Ps} The maximum value of the term in 6 proved small ( < 0, 003) for 40 MeV 165 bombardment of 14a4 (8 = 0,018), and was corre- spondingly smaller for bombardment by lighter ions, The energy loss of the recoiling nucleus in the target effectively reduces the value of 8 to below that calculated for a freely recoiling nucleus, <A calculation of the range of 17 MeV 11404 ions (8 = 0, 018) in !4¢q using the formulae of Lindhard, 70 et al,, (1963) yielded a value of 200 ugm/ em”. This range, probably accurate to + 50%, is less than the thickness of any of the li4a, or 216 average energy of 8 MeV is assumed over the range of the recoiling nucleus, the time in which the nucleus comes to rest is of the order of 10°! second, which is much shorter than the lifetime of the excited nucleus, Therefore, no correction for nuclear recoil Cd targets used in the cxperiments, If an was applied to the gamma-ray angular distributions for the cadmium nuclei, In the tellurium measurements however, in which the targets were only 80 pgm/ em? thick, a correction to the gamma- ray angular distribution for nuclear recoil was applied. Since, in this case, most of the gamma-ray de-excitations for 165 ion bom- bardment took place in a region just behind the target, a slight 16 difference in relative efficiency for te ion and ~’O ion bombardment resulted, The corrections to the yield for this were less than 0, 2%. 10, 11, 12, 71 REFERENCES Alder, K,, Bohr, A., Huus, T. Mottelson, B, and Winther, A., Rev. Mod, Phys, 28, 432 (1956). Alder, K, and Winther, W., Proceedings of the Symposium on "Recent Progress in Nuclear Physics with Tandems", Heidelberg (1966). Bohr, A. and Mottelson, B,, Kgl, Danske Videnskab. Selskab Mat. fys. Medd, 27 (1953). Breit, G. and Lazarus, J. P., Phys. Rev. 100, 942 (1955). Cookson, J. A, and Darcey, W., Nuclear Physics 62, 326 (1965). Dang, G. D., Dreizler, R., Klein, A, and Wu, C. S., Phys. Rev. Letters 17, 709 (1966). Davydov, A. S. and Fillipov, G, T., Nuclear Physics 8, 237 (1958). de Boer, J., Stoksiad, R. G., Symons, G. D. and Winther, A., Bull, Am. Phys. Soc. 9, 718 (1964). | de. Boer, J,, Stokstad, R. G., Symons, G, D. and Winther, A., Phys. Rev. Letters 14, 564 (1965). Douglas, A. C., Atomic Weapons Research Establishment Report NR/P-2/62, Aldermaston, England (1962), Douglas, A. C., private communication (1966), Eichler, J., Phys, Rev. 133, B1162 (1964), 13. 14, 15, 16. 17, 18, 19. 20, 21, 22, 23. 24, 25. 72 Gangrskii, Yu, P, and Lemberg, I, Kh., JETP 15, 711 (1962), Hansen, O. and Nathan, O,, Nuclear Physics 42, 197 (1963). Herskind, B, and Yoshizawa, Y., Nuclear Inst. and Methods 27, 104 (1964), Lindhard, J., Scharff, M., Schott, H. E., Kgl, Danske Videnskab, Selskab Mat. fys. Medd. 33 (1963). Lindquist, T. and Marklund, L, Nuclear Physics 4, 189 (1957). Litherland, A, E., Clark, M. A. and Broude, C., Phys. Letters 3, 204 (1963). MacDonald, N., Nuclear Physics 48, 500 (1963), MacDonald, N., Phys, Letters 10, 334 (1964), Marion, J. B,, Nuclear Data Tables, Part 3, Nuclear Reaction Graphs, National Academy of Sciences, Washington, D. C., (1960). Masso, J. F, andLin, D, L., Phys. Rev. 140, B1182 (1965), - Mayer, M. G,.and Jensen, J, H.D., Elementary Theory of Nuclear Shell Structure (John Wiley and Sons, Inc., New York), (1955). McGowan, F, K., Robinson, R. L., Stelson, P. H, and Ford, Jr., J.L,C,, Nuclear Physics 66, 97 (1965). Miller, R. G., Ph.D. Thesis, California Institute of Technology (1966), — 26. 27, 28, 29, 30, 31, 32, 33. 34, 35. 36, 37, 38, 73 Pearson, J. D., Ph.D. Thesis, California Institute of Technology (1963). Perlman, M. L, and Welker, J. P., Phys. Rev. 95, 133 (1954). Rose, M. E,, Phys. Rev. 91, 610 (1953). Simpson, J. J., Eccleshall, D., Yates, M.J.L. and Freeman, N. J., to be published. Stelson, P. H, and McGowan, F. K., Phys. Rev. 110, 489 (1958), ‘Stelson, P, H, and McGowan, F. K., Phys. Rev. 121, 209 (1961). Stelson, P. H., Milner, T. W., Ford, Jr., J. L,C., McGowan, F. K,, and Robinson, R. L., Bull, Am, Phys. Soc. 10, 427 (1965). Tamura, T. and Ugadawa, T., Phys. Rev. Letters 15, 765 (1965). | Tamura, T, and Ugadawa, T., Phys. Rev. 150, 783 (1966). Temmer, G. M, and Heydenburg, N. P., Phys, Rev. 104, 967 (1956). Wiedling, T. and Asplund, 1, Arkiv Fysik 18, 65 (1960). Winther, A. and de Boer, J., 1965, A Computer Program for Multiple Coulomb Excitation, Technical Report, November 18, 1965, California Institute of Technology. Winther, A., private communication (1966). 14, TABLE 1 a4 em”) used in the computer The matrix M,,, (in units of e- 10° calculations, (u,_.)* = B(E2,s-1r) x (21, + 1). My and Mo. were fitted to the experimental dala, and the Mig values for the higher transitions were taken from McGowan, et al., (1965). The assign- © ments of the indices r and s are indicated in Fig, 11. Since no measured value for Mog was available for 1144, this matrix element was taken to have the single particle value. . S| 4 2 3 4 5 6 7 | — A4og 1 0 My, 0 40,09 0 0 +0,09 2 | My, Mgg -0.31 -0,83 -1.35 -0,13 -0,31 3 0 -0, 31 0 0 0 | 0 0 4 | 40,09 -0.83 0 0 0 0 0 5 0 -1.35 0 0 0 Oo 0 6 0 -013 0 0 0 0 0 7 | 40.09 -031 0 0 0 0 0 : | 11604 1 0 Myo ~ 40.147 0 0 2 | My My, -0.66 -1,32 -0,31. 3 | 40,147 -0,66 0 0 0 4 0 ~1.32 0 0 0 5 0 -0.31 0 0 0 The matrix M,,. (in units of e+ 10° 15 24 TABLE 2 cm”) used in the computer calculations. (M,.)° = B(E2, sr) x (21+ 1). Mj. and My. were fitted to the experimental data. The assignments of the indices r and s are indicated in Fig. 12. The derivation of the values of My 4> Mog and Mo4 is detailed in Appendix II. B. Ss r 1 2 3 4 126,,, 1 0 Myo 0 +0, 07 2 Mo, Moo -1,10 -1. 00 3 0 ~1,10 0 0 4 +0, 07 -1, 00 0 0 128,56 1 0 Mj 0 +0, 07 2 Mo Moo -0, 94 ~0, 20 3 (0 ~0, 94 0 0 4 +0, 07 -0, 90 ) 0 76 TABLE 3 Experimental values of P= doo / BI 1 do; measured at a mean labora- tory angle of 165°, E is the effective bombarding energy. The number in parentheses is the power of 10 by which the value of Pox p and its assigned error should be multiplied. Errors are random only. The data obtained at tHe energies greater than 10 MeV have not been incorporated in the analysis. 114, Projectile EMev Poxp 165 46, 03 0, 250 0, 134 + 0, 002 (0) 40.80. 0, 300 0.868 + 0.009 (-1) 40,45 0,304 0,828 + 0,010 (-1) 40, 05 0,308 0, 802 + 0, 007 (- 1) 36, 85 0,350 0.552 + 0,007 (-1) 36.10 - 0,361 0,505 + 0,006 (-1) 34, 85 0,381 0.413 + 0,007 (+1). 33, 73 0, 400 0,362 + 0,005 (-1) 33, 00 0,414 0,331 + 0, 004 (-1) 120 30, 04 0.300 0, 561 + 0, 006 (-1) 27.14 0,350 0,356 + 0, 004 (-1) 24, 86: 0.400 0, 226 + 0,003 (-1) © tue si 0.300 0.775 + 0,012 (-2). 9,74 0,302 0,781 + 0,013 (-2) 9. 71 0. 304 0,757 + 0, 008 (-2) 8.78 0,354 0,472 + 0,006 (-2) 8.05. 0. 406 0.307 + 0.005 (-2) continued 77 TABLE 8 (contimed) Projectile E 5 P MeV exp 11.30. 0, 240 0,188 + 0,002 (1) 10,73 0, 260 0, 120 + 0, 002 (-1) 11604 165 41.63 0, 269 0, 110 + 0, 002 (0) : 37,65 0,313 0,709 + 0,012 (-1) 34, 55 0.356 0.491 + 0,007 (-1) 12 30.66 - 0,269 0,705 = 0, 009 (-1) 27.73 0,313 0,457 + 0, 006 (-1) 25. 46 0, 356 0,310 + 0, 005 (-1) tHe 9.966 0, 269 0, 102 + 0, 001 (-1) 9, 019 0,314 0,659 + 0, 014 (-2) 8,304 0.356 0,445 + 0,009 (-2) 12, 08 0, 200 0, 211 + 0,004 (-1) 11.35 0. 220 0.172 + 0,003 (1) 10.73 0, 240 0, 131 + 0, 002 (-1) 78 TABLE 4 Experimental values of P = do,/ z do, for 126.4, and 128.56 measured ata mean laboratory angle of 165° —«E is the effective bombarding energy. The number in parentheses is the power of 10 by which the value of Poxp and its assigned error should r multiplied, Errors are random only. The data obtained at “He energies greater than 10.5 MeV have not been incorporated in the analysis. Projectile EMeV § Poxp 126.4 165 43, 21 0. 352 0.608 + 0.006: (-1) 39. 56 0, 403 0.3954 0,007 (-1) 36.61 0. 453 0,268 + 0.003 (-1) tHe 10, 44 0, 352 0.5124 0.006 (-2) 9,58 0.403 0.3284 0,005 (-2) 9,15 0, 433 0.261+ 0.005 (-2) 8, 89 0. 453 0.2194 0,004 (-2) 11,54 0, 302 0.804% 0.008 (-2) 11, 06 0.322 °° 0,662+0.013 (-2) eontinued 79 TABLE 4 (cont. ) Projectile EMeV E P xp 128, 16, 42,96 0, 396 0.4374 0.006 (1) 41, 60 0, 416 0.357+ 0.004 (-1) 39,75 0, 446 0,289 0,003 (-1) 37, 08 0, 495 0,193+ 0.002 (-1) tHe 10. 11 0, 416 0. 296 + 0.004. (-2) 9,67 0,446 » 0.2294 0,004 (-2) 9, 042 0, 495 0,156 0.002 (-2) 11, 58 0.337 - 0,57640,018 (-2) 10. 60 0.386 0.3774 0.004 (-2) 80 2apIO SAT "SOSTS [[@ Ul OGL "O- 2g 0} uaHE} Sem aly pue pajou Anpyidxe azaym ydaoxe | apqey wor; use} d19M SPSUIIS XPAWEW ap Jo Sapnyyusew syT -(gz) ‘ba 0} Suppo." Aroayy uvoyjyeqanzzad Jaepszo jsAayj BujSn payemayes oram d JO Sanfea sux ‘Z < | XOPUT YEA STaAaq TT? Bup}ayZeu payndwos aram TeAay O89 5 16 Sante ou *sseu arefoad ayy uo puadap yuautour atodnipenb o7}e}s 973 Jo $}0937a ay} MOY |JEL]SUOUAp SOUT] INOJ 4Se] OYL “W19} BdUS1apIO pT] Tapsz0 ySanojq ££600°0 96800'0 Oo06cd"o | (9-)TTT % (8-)sog’s (9-)z8e'E (9-)oTa"’s «= (u-) S42" (e-)ort't «= (tezz} ove - - - O'T+ | 06%'0 | OOF ££600°0 82600°0 zeeo0"o f+ (9-)oso’e [e-)agn"c (9-)zez*r (9-)rzL ts = (L~)zze-e (€-)OST"E (I-2ez't | oO - - - 00 06%°0 | O0T i 6F9~0"o §=—sT9S0°O §=—s950°O | (S-)eea tT (9-)pet‘z (S-)oze"L (S-)eeg’s = (S-)oRL'T (e-)ope’9 = (E-)ORT'T | O'O) - - O'T+ toe’o | 00 6F90°O | Fz90°O §—« ZOO | (S-)OTA*T (9-)eet"e (S-)oaz"2 — (S-)ebE’s (SG -)eRL'T (€-)109°h —(E jeett | 00 - - - 00 Tof"o | OOF 221 s660'O st80°0 czgo’o | (S-)o9a"e (9-)zto’s (y-)eL9'T (b-)te0l = (S-)060°F (€-)kSL°6 = (T-)680'T | O'O + - + O'T+ | 606°0 | 0'OF S660'0 £180°O 08z0°o | (S-)ezo's (9-)ete'e (P-)zpo't (%-)S00°I — (S-)800°F (-)s12'6 = (Ie60'T ] OO - + + O'T+ 60f "0 | OOF s660°0 t80°0 68200 | (s-)ees"z (9-)ene"> —(F-)ISOT (PTOI. ~—(S-)TE0°F (e-)s6£°6 (Te6o'T | o'oO - - + O'T+ | 606°0 | 0‘OF se60°0 ctso"O otg0°o | (S-)ogo’s = (9-)see"h (F-)P99'T (b-)zs0°1 = (S-)990°F (g-)ee9°6 = (T-)060'E | OO - + ~ O't+ | 60€°0 | 0‘OF se6e0’o f190°0 8g0'o | (s-jo9s‘e (9-)9z0's (B-)eTE*r «= (b-)S80°T §=—s (S-) OTF (€-)18L°6 = (T-680'T | Ott - - - OTt 3060 | 0 ‘OF s660°0 St60"0 %60'O | (G-)beS ‘2 (9-)e96"R (e-)egg"t (S-)L08°s = (S-) OT > (8-)IZE'T = (T-)PLO'T | OO - - 00 3060 | 0‘OF $660°0 450T‘0 990T0 | (S-)sez‘z (9-)uze" (Jagat = (S-)eso's —(S-)oLO"® «= (z-)g9zE = (T-Jo90'T | ovo - - - ot- | soe’o | 0'oF - 8Lt0°O §«czeo"O )§= ezeo’o | (s-)eat't = (u-)eTZ"@ «= (G-)ezo’e = (S-)tee’s = (9-) teS°8 (e-)seu"S = (T7689 | OO - - ~ Ott | ctp'o | o'ce 6650°0 T0S0;0 oTso'O | (S-)oOe TE (9-)uzo'e (S-)ooe’s9 (S-)ese’r (S-)ztB'T (e-)sep"L = (t-)eee"T | OO) - - - O't+ | 296°0 | 098 s660°O0 ete0°0 ozeo"o | (S-)ese-e (9-)zt0’s (b-)ezg°t (¥-)IG0T = (S-)080°F (g-)1s2°6 = (T-)880'T | O70 + - - O't+ | 608°0 | O'OF OP8T'O GErT'O FtbT*O | (S-)ose°o = (S-)ethE «= (F-)OTO™® «= (H-)oTL'2 «= (B-) BOT (2-)eLz"t = (Z-1969'L | O'O - - 7 O'T+ 0s2°o | 0'9F % aapio 12AaT bop %Qp Sop Yop fop 2op lop Ty Py LT Fy Be yy SIs omy dusod di (u9j Jo Suamod ajouap s3ay9%I1g UT Sraquinu ‘ueTpEra}s /sureq) (m9 pz-OF * 3) sjuawaya AN am suo}}Ies Sso1d L10}e210qeT XJajyeu Jo SanyvA Jo sudig 2 Asa | -daforg sap nqeqosd wuoyyeyoxa-aq MOYS SIU] ING] SAYS GUL zayndwos ayy duysn pouyrygo 989 30 ap3ur As0jye20qQ¢] & Je pa —n °3 TE “OTs Ut UMoYs se Pit PIT S STaVL 34} OF agnqyazquod ‘Arcayy uoyeqanzsad up ‘YoTYM syuauraya xqAyEUI aSOYZ Jo SUITS aaq}ETEI ay} UC pue Fy Uo aduapuadap ayz ayes} -SO] IF SOUP] UaASS Jxou BY “JUaUIpIequIcg Op, IO} SITQeqoid uojj{TIIxXa-ap pue suctpas sso19 ay} Jo aduapusdap Adsaua ayy Uy STAAI] Of AIjar yO} T SIdTPUT IYL ‘J XPpuaddy uy paqisosap wesi01d TO} d SaWTMEGOId uoWePOX|-ap pus SuOTIIaS S3019 payee; 81. PY GG LT. on G gto .2-01 °® SW 89h °0- = ry) ral ‘ ‘ 9] e = - pit) 9, 0F *29 oC W) = (2 - {0 “ead 6° (8T°O)ST’0 FLO‘T+ (£0 '0)S00 ‘0 + 88h °0- + Py ° * = e * a — en - é é L°t (LT°O)PT'O F6E*T+ (£0 '0)S00 ‘oO + 88h °0 6 OH, O27 Og, oy | | Poort Ph (9°0) b'O * “T+ §«©(S0O)6TO‘O FLBL*0- = CC Oy 02 (61°0)L0'0 FLL°0+ (0 °0) £00 ‘0 F 8PL °0- + + tz (61 °0)L0'0 F98°0+ (0 °0)800 ‘0 F 8PL “O- - + (4d (61 °0)L0'0 FOO°ET+- ($0 °D)$00 ‘0 + 8PL ‘0- + - ve (61°0)L0'°0 FOT’E+ (80 °0)€00 0 + 6bL °O- “ - LT oH, “Oey ‘Oo; POSIT : 183 §9) ° wo 9 paarenbs go pe-FF 9 2 pe-7F “Ty Vg dxa saTtpoforg “WD ae aly jo SUSIg | "SOILS [2}O} OY} 1B SasayjUered Ul *eyep ay} SUTPT] UL pepn{oUT arom S10119 WopueZ ATUG axa Aq peyousep St I} & JoF pasn syzutod TeyUowTIodxe jo Tequinu oy, “sAtyesou uosoyo ATIreaziqase St eh jo USTS OYL “T aTqVy, WOI] Udye} ore SyUSUIETA XIAyeW ZY IOYJO ay} Jo SonTeA OUT, "¢E OTqRL UT UATE dxo . sotjquenb parmmsveur oy} 0} Coy pue aly SJUsuIaTe xtazyeur au} Jo PW} sorenbs-jsvayq 9 WIGViL 82 TABLE 7 Results of a least-squares fit of the matrix elements Myo and Mo to the measured quantities Pox p given in Table 4, The values of the other matrix elements are taken from Table 2, The sign of Mio is arbitrarily chosen negative, Only random errors were used in the least-squares fit to the data, In parentheses are the total errors, In all cases the number of experimental points used in the least- Squares fit was 7. Sign of Mi9 Moo Chik M -24 2 -24 2 14 (e- 10 “* cm‘) (e- 10 “~ em“) Squared 1267, - +=0,698+ 0,004(0.025) +0,53+0.10(0.13) 87 + =0,697+4 0,004(0.025) +0,32+0,10(0.11) 6.5 128,,, - =0,625 4 0,004(0,023) +0,35+0,09(0.17) 92 + =0,624+ 0,004(0,023) +0,14+0.09(0.13) 38.3 e” . 10748 em* 2 B(E2, 0° + 2") = (M, 4) , _ -24 2 Q, = -0, 758 M,. e+ 10 “" cm FIGURE 1 Tlustration of the terms which contribute to the excitation of state f from state i. The first and second order excitations are denoted by double arrows and the interference terms by single arrows. (Page 12) 84 je fot joj<! jo 2s-| jo 2e| $=! (2? (z2)¢ (2? zz? 0 + y VV \Y 2 WT) A\ + \Y FIGURE 1 85 FIGURE 2 The quantity K(§ ve 6) defined in (25) plotted as a function of E fi" The small slope of the curves indicates the weak dependence of the function p on the bombarding energy. (Page 15) 86 O01 60 80 40 90 GO 0 P| 3 46) cO qT q qT L i 009 006 T (@°'43) » FIGURE 2 87 FIGURE 3 The interaction constant Xs defined in (12) calculated for the excitation of the first 2° state in 144 energy range from 0,1 to 1.0 times the Coulomb barrier and were calculated using a value of B(E2, o* + 2°) = 0,50 e” - 10 4 em”. (Page 18) Cd, The curves span an -48 88 (A9N) ADYSNA JILOACONd Ol 09 oe 01 | O¢ I T , irs fini J PD 4 Yos|2— 'x| i 7 1 l fexe) +£0'0 Liu | i oS 8 oO LOVUYSLNI 0 Oo ; |=—'x| invisnoo No! | © o Ot FIGURE 3 89 . FIGURE 4 The arrangement of the detectors and beam collimation system within the target chamber (side view). The arrow indicates the direction of the beam, The items numbered (1) through (5) comprise the beam collimation system, (1), (3) and (5) denote tantalum collimators with inside diameters of 1.5, 1.8 and 2.0 mm, respectively. A lead spacer (2) minimized background gamma, radiation from the first tantalum collimator (1). Collimators (3) and (5), separated by an aluminum spacer (4) prevented the beam scattered from (1) from striking the back- side of the solid state detector (6). The sensitive surface of the solid state detector had outside and inside diameters of 10 and 4mm, respectively. The target (7) was placed about one centimeter from the sensitive area of the solid state detector. The face of the NaI detector (8) was 3.3 cm from the target and the beam stop (not shown) was about two meters from the target, (Page 21) | $0 Wd vl ol Ol er om i 1 i] j | Wr “IsskzZA 299¢¢0¢e¢ 6 1 FIGURE 4 91 FIGURE 5 Block diagram of the electronic apparatus. (Page 22) solid state particle detector preamplifier wide band amplifier linear amplifier single channel analyzer 92 INOL gamma detector, photomultiplier preamplifier fast coincidence time - to- pulse- height slow coincidence wide band amplifier linear amplifier single channel analyzer ; gate 400 channel onalyzer FIGURE 5 93 FIGURE 6 114 16 Cd by O ions. The window used to gate the time-to-pulse-height ‘Gamma-ray spectra from the bombardment of spectrum is indicated, (Page 23) 94 YAEWOAN TANNVHS 002 Osi ool OS ee A bd re a rt! TTY re T - So e @@ ® 9 e oe ”y eee @ eee ~ L MOGNIM / ee & 088 @e esoe e | ee ef ee | A®W 8SS0 | - e - 4 5 oe so ee 4 Ss @ e —~ - ° e ee 4 — ° e —! ° SI TONIS L e- e 5 ; 4 5 e vy - z e L be ® = a e_] SIN3GIONIOD e e oe é e ee e r e ° % 4 om | = WASAYD ION "WS Zg°Z x "wo Zon ° ® 4 | AW LOp= 3 | . 7 i (Bulyo0q uoqo2) PI p14) e 6 a 5 VuLoadS VNWV9 8 -] a : e ® - be e ~ G 1 1 L [ | 4 1 | ii j f 1 v if J I it | AL I l = ° oo! ooo! SJONAIGIONION JO YSEWNN FIGURE 6 95 FIGURE 7 Gamma-ray spectra from the bombardment of 12876 by 165 ions. The window used to gate the time-to- pulse-height spectrum is indicated, (Page 23) Y3SEWNN TANNVHOD ost ool os 0 i 1 = © e ome ee ae z ee e = °° ee e v e ee © - 2 e 9 e@ e - e e . 8 ee - mn ’ Ol e eaoe @ e e e e e bd e ° e °. D) > Sa ae r ° e etre 02 oO 3ON3GIONIOQO—~ ote *sse oh 8 oO e e a e ¢? MOGNIM ° ' ov t ' : f e e 09 = e e 038 = oOo! % 3 a“ e : e TWLSAYMD ION WO ZoZ x ws poe a AWELip= 3 002 (Gu1490q Uoquod) al,,, VuLosdS VNWVS L | 1 00¢ APW ChZO i SFIONIS 009 a 008 4 yy gp yg f gp gy gy pt gg gg lg oool SSONSGIONIOO 40 YSEWNN FIGURE 7 Q7 FIGURE 8 Spectrum of 16 ions scattered from 11464 in the angular range 155° < @< 175°, The elastic and inelastic groups are not resolved, The window used to gate the time-to-pulse-height spectrum is indicated, (Page 23) 98 YASWNN TANNVHOD 002 O08! 091 Ovi OZ] fore) | Os 09 Op 0g | T q J i] T T T T T ry e ° 0% ee e 7 0% See, o° ° of. e 4 *: a | é % ® é 309°? a && e _°@ MOQNIM | “eee? e Boe ue, | “ ® e o | e oo ~ e @,° r) r) 4 r’ ° e e r e e f 4 e e e e . 7 e é | . ff 4 - 9, - e £ oSLl> @ > oSGI Ya? AaW be = ly (6u1y90q woqs09) POsn WNYLOAdS JTDILYVd Og, 4 i I J l | L Ol 02 Ov 09 08 00} ° ° N SLNNOOD OOP 009 008 0ool 0002 000b FIGURE 8 99 FIGURE 9 . spectrum of 16, ions scattered from 128... in the angular range 155° <a < 175°, The elastic and inelastic groups are not resolved, The window used to gate the time-to-pulse- height spectrum is indicated. (Page 23) 100 Ode YASWNN TANNVHD 002 ‘8! 09! Obl 02! ~—s 001 08 09 Ov oz O.. haan | T T T T T T T Ol e e °® 8,0 eo -~ " fo bet, ° oe , 7102 20, te Bee ‘ *, e ¥ @& e ° , gen e eo e . + 08 ¢ = ?*. —{ 00] @ _ * MOGNIM | a L ry - e ° ° + 002 e 6 . ° 4 oop : * e + 009 = @ a oos — Pa) — 0001 e @ e e -+{ 0002 ¥ | oSLi5 Mg 5 .gcI a MW 96zb=°%3Z = ogOF (Buryonq uoqins) ®1oz r WNYLOAdS FIDLYVd Og, 7 9009 . + 0008 SLNNOS FIGURE 9 101 FIGURE 10 Typical time-to-pulse-height spectrum; the spectra obtained in the bombardment of 11404, 11604, 126 16 and 128.16 were all similar. The small slope in the randoms background is due ‘ toa slight asymmetry in the timing pulses. The FWHM corre- sponds to a time resolution of 4.7 nanoseconds; the yield was counted over the base-width of about 22 nanoseconds. The subtraction of randoms background amounted typically to about 2% of the peak. (Page 23) 1000 COUNTS 800 600 400 200 100 80 60 40 20 oO oO 102 TIME OVERLAP (ns) 15 30 45 60 75 90 a en | { ! t | | | & _ e@ * TIME-TO-PULSE-HEIGHT - a SPECTRUM . — 0-2 20 40 60 80 100 120 CHANNEL NUMBER FIGURE 10 103. FIGURE 11 11 116 4Cq and included in the analysis are indicated by arrows. The | Level schemes of Cd. The matrix elements M,., values of M,,, ave given in Table 1. (Page 24) 104 Pon | eye) iSO 2z21<— 28el (A®W) ADYSN3 +90 NidS Poon esso eel eb 802.1 £ b 8c SO¢ S £9¢'l XSONI = (A®W)ASYSNA | 2 2 o ¢ wz Ob b 86S Moen) a | NidS X3QNI FIGURE 11 105 © FIGURE 12 12674 and 128 4 The energies and spins Level schemes of are taken from Cookson and Darcey, (1965). The matrix elements M5 included in the analysis are indicated by arrows and are given in Table 2. (Page 24) - 106 82l ) 40 Zepzio HY rl i 2 22S) “(2) (A®W) ADYANI NidS XJQNI pL 4,29 1 oz g9990 O9¢l any cov il (A9N) ADYSANS I Oe v) ¢ OUD NidS XJQNI FIGURE 12 107 FIGURE 13 Spectra of He ions scattered from 11404 in the angular range 155° < @ < 175°, The position of the window used to gate the time-to-pulse-height spectrum is indicated. The three large peaks in the singles spectrum are due to scattering from cadmium, oxygen, and carbon, respectively, The smaller peaks are due to 446 ions which have lost some of their encrgy in passing through the dead layers on the surface edges of the sensitive area of the counter. The coincidence spectrum shows a displacement relative to the singles spectrum due to the lower energy of the inelastically scattered projectiles. A small peak in the coincidence spectrum due to random coincidences appears at the energy of the elastic peak, The peak-to-tail ratios for the singles and coincidence spectra are seen to be the same, indicating that all the counts above the window are due to scattering from cadmium, NUMBER OF COINCIDENCES 1000 800 600 400 200 80 60 40 20 108 100 l T T Sie PARTICLE SPECTRA "Seq (carbon: backing) E, =9.8 MeV He 155° = 8.4, = 175° l | | COINCIDENCE —= SINGLES —————— WINDOW | | L 20 40 60 80 100 120 140 160 isd CHANNEL NUMBER FIGURE 13°" 200 109 FIGURE 14 Comparison of the experimental quantities P = do,/Z,do. for 4. 12 16 114 aii He, ~C, and ~ O bombardment of Cd with those calculated in first order perturbation theory. The deviations from unity are attributed mainly to the quadrupole moment of the first ot state. The three solid lines represent the results of a computer calculation in which the two parameters Myo and Moo were varied to obtain the best fit to the experimental data. Pes ast order calculated using the fitted value of Myo. The dotted points were was not included in the analysis, The experimental errors and those quoted for M,. and M,, are random only. (Pages 29, 35) 110 LU t q "46g Pp p $ ‘he FULL CURVES ARE LEAST-SQUARES FIT FOR ——!"2— exp. 4 \%¢ . frat order 4 Peieat order ry FITTED VALUES (M,4,M,p-vel' B(E2,0* © 2+) * 0.860 + 0.008 @.107*%cm 4 "0 Q_ *-0,83 0.05 ¢:10724 em? 1LOo-— : —) ee CS EY LE SS A MeN CANN SONNE cle T l | . i 25 27 30, 33 36 - 40 46 aed =4 10ON ENERGI (MeV “He > 169 Es ) 1 A 1 aso 0.35 0.50 FIGURE 14 O25 35 30 85 1S 111 FIGURE 15, 116 Analogue of Figure 14 for “Cd. (Pages 29, 35) 112 os” os sé oO 02° sz o¢ Pon se" VA) | Og)! r iv T ze + S310¥3N34 2, o¢ 22 $2 nor laHy 21 " ol Tr H Hi . £ 4 $ $ | i i a i — vT —f ‘ 4 ? —_— — i : F 2%42-01-® O1'0 # SOs #0 ; Ont eon | 95,01 29 BOO'D ¥ 1Z9'0+(42 = 40°23) 8 +(0A— WW) SIMWA ILLS d y Se 2 sep0 mu, + + Ja é ee away Od Ld SauvNOS-1sv31 3uv S3AuND TNS a“, § oe so - 06 sé ot FIGURE 15 113 FIGURE 16 Comparison of the experimental quantities P for 4 He and 16, 126 exp Te with those calculated from first order bombardment of perturbation theory. The deviations from unity are attributed . mainly to the quadrupole moment of the first 2° state. The solid lines represent the results of a computer calculation in which the two parameters Mo and Moo were varied to fit the experi- mental data, Prins t order WS calculated using the fitted value of Myo: The dotted “He points were not included in the analysis. ‘The experimental errors and those quoted for M1 and Mo are random only. (Pages 29, 35) 114 o¢0 seo Ovo svo i | 1 I Og; —_> , bP ov Ov se 9¢ (A®W) S3ISY3SN3 NOI ie Ol 6 O60 }— ° . —; 060 t 001 +- : | t — 001 r 2u9 4,01: 2 20°20b0-= 75 : 7 p49 g,O1-29 SOO F280 = (,2-+-,0'2a) a: @0A-"W) SAMA C314 spun, ; Og, Y 4ep0 wy, wes, NOS Lid S3UVNDS 1SV31 3YV S3ANNd TINS oH, F dea SL gz FIGURE 16. 115 FIGURE 17 128 Analogue of Figure 16 for “" Te. (Pages 29, 35) 116 oOo] seo” ovo | S¥O oso | t q U Og; —> bry ap Ov SE 9¢ (A9W) SAISY3N3 . NO! 5H, > iI Ol 6 I 4 reese —< {ol = T ; — L °) 4 [ 1 I a TO — “I t 4 rm suo 4201-2 207 L20-= 7 . 4 pX? 9-01-22 SOO F 162’0 = (,2-+-,0'23)8 (2A -"W) SANWA G3LLIJ AOPsO 484)} hy Og) t vepi0 Wily Fuss, 404 Lid S3YVNOS LSV3T1 Juv S3ANND TINJ °H, } dxt, *L921 I I I ! 060 FIGURE 17 001 117 FIGURE 18 The data and least-squares fit of Figure 14 replotted but now normalized to Pomp (Q, = 0), the latter being the value of P computed with Moo = Q5 = 0 and with all other matrix elements the same as before. Here the dispersion between curves corresponding to the different bombarding ions is due entirely to a non-zero quadrupole moment of the first 2” state. (Page 37) 118 $20 . o¢'o seo Ov'0 I raw) [09 9% Op o¢ ee 6319YINI{ 95, o¢ 22. G2 NOI H, I ol 6 @ seo os { J J 4s30 Dt it I ] l — I ; : | jt; J O60F- i ' 060 : , S60} | t +s6'0 —~ | | j 4 001 3 1—00't z942z-01'? SO'0 F29'0- = %0 * . ‘ 2) Oo , p22 gp-Ol- 2? SOOO F09S'O+(4Z—40'Za)a(eA-“"WN) S3NIWA OLLI af (0-0), 22 dwos i) rane YO4 Ld S3YvNOS-1SW31 3yv S3ANND THNJ “ ae $ PI vi FIGURE 18 119 | FIGURE 19 116 Analogue of Figure 18 for Cd, (Page 377) 120 02°0 S20 o¢’0 seo | SHOUINI< Da, of ee se not eH, 2t T ol 6 ss'o- . +s8'0 T : , ! po i | i ; i 4 -o6'ol- ; +06'0 ' $ ? ‘ ‘ 4 SéOr- —_—_- — _. _ _ | S60 oot 007 24,2-01-@ OO FSO1- =% PS pp -O1- 2? 8000 F1Z9°0 = (42 40 '23)8 (90-"W) S3nWA GILLIS 5 (0=20/"°y (0=%0) 4 zi axe ames YO4 Ll4d S3YWNOS-1SW3] 3uv S3ANND TNs OM d PQon , ! J J 1 FIGURE 19 . 121 FIGURE 20 The data and least-squares fit of Figure 16 but now normalized to Peomp'2e = 0), the latter being the value of P computed with Moyo = Q, = 0 and with all other matrix elements the same as before. Here the dispersion between curves corresponding to different bombarding ions is due entirely to a non-zero quadrupole moment of the first 2” state. (Page 37) 122 O60 001 o¢o sco ovo Svo T ] T Og) —> bb 2p Op ge 9¢ (A2N) S3I9Y3N3 NO! eH, — ii Ol 6 P i a EEE i a r {uo 4201 ® 20° 0v0-= 7D | 7 pW g,-Ol-29 GOO’ F8¥0 = (,2 = ,0'29) a :(24-"W) SINTWA G31LIS (0 =%)°Y Oy ¥ — (0=70)Yy ——~“gaza, Od Lid S3uVNDS 1S¥31 Juv S3AUND TiNd Hy | axe, 192) 060 001 FIGURE 20 123 FIGURE 21 128 Analogue of Figure 20 for ““ Te. (Page 377) 124 ool Seo ovo Sto os‘0 T Og, —> {(A9N) S3ISY3N3 NO} i > ‘ fm oe ee oe oy r 2W242 01-8 L205 220-= % (0 = 2H) 4% duioay bb ob Ol ‘Op Be At 1 ee! i yp? g,-Ol -29 GOOF I6€°0 = (,2 = ,0'23)8: (24-"W) SaNIWA G3LIId 403 Lid SAYVNOS LSV3T 3YV SSAYND WMN4 oH Oy ¥ H, (0=20) — as mm i potipp piri ii tin 060 007 FIGURE 21 125 FIGURE 22 Systematic variation of some properties of the cadmium and tellurium isotopes. In each case the maximum and minimum values of My o/ | M,4| are plotted (corresponding to both signs of the interference term Po +_ ort ot ). The energy and transition moment ratios have been obtained from Cookson and Darcey (1965) and McGowan, et al., (1965). The errors for the cadmium B(E2)-ratios are typically 20% while those for the corresponding values in tellurium are probably 30% to 40%. (Page 51) . 2.4 22 20 1.8 Lor 1.4 2 1.0 0.8 06 126 - oneal |_.-PHONON. MODEL OL ‘ B(E2,4*+ 2") a é B(E2,2°-0") ° B(E2,2*=0") Ey+ E2* Ey'* Ep+ x x x xo B(E2,2’%»2*) ° Ooo ee ee ee 62 64 66 68 70 72 74 76 78 WO i214 22, 124126 128130, ' Cd Te 2°48. Z*52 FIGURE 22 1.8 1.6 14 12 1.0 0.8 on IM,e | +0.2 0,0 a ir ro 127 FIGURE 23 Momentum spectrum of elastically and inelastically scattered ions at a laboratory angle of 150° following bombardment of a thin 11404 target by 9.76 MeV 4He ions. This spectrum was recorded in four overlapping measurements with an array of 16 counters in the focal plane of the spectrometer. (Page 60) NUMBER OF COUNTS 350 300 250 200 150 i00 50 128 NMR FREQUENCY (Mc/s) FIGURE 23 T T T Giab * iSO” > xe 100 "4¢4 (0.558) . l4eg (0) NN 4 , « q — — q ? + } -* 4 ral oe, f § S Ye “ .¥ "Fae a. 26.0 26.5 27.0 275 129 FIGURE 24 Momentum spectrum of elastically and inelastically scattered ions at a laboratory angle of 150° following bombardment of a thin 114 recorded in two overlapping measurements by an array of 16 — counters in the focal plane of the spectrometer... (Page 60) Cd target by 40.7 MeV 165 ions. This spectrum was NUMBER OF COUNTS 130 { i i if i | so "4Cq(0) 350 9 00 El6 = 40.7 MeV 3 Bigb = 150 250 + 200}+- 150 - 100 + aa 114.¢4(0,558) DAW, | 298 300 302 304 306 30.8 NMR FREQUENCY (Mc/s) FIGURE 24 131 FIGURE 25 Gamma-ray spectra from the bombardment of 1267, by 42 MeV 16, ions. The spectra were taken with a 20 cc lithium-drifted germanium detector. The coincidence spectrum was gated bya window placed on the 667 keV 2* oF transition photopeak recorded by a 10. 2-cm-long x 12, 7-cm-diameter NaI detector. The background consists almost entirely of real coincidences whose source was most likely the 12, (165 p «) Na reaction due to carbon accumulation on the target. The arrow indicates a coincidence peak from the 2° + 0* transition and the horizontal bar indicates the expected position of the 4" at gamma ray. (Page 62) 132 02 ” Op @ 09 m 03 Oo OO! C O2l Ss a Obl og MAGNNAN TANNVHO OSE ss1oNis—” im HA i | rnanowee 7 | ! i AVAdOLOHd 2 ~—0 dO NOID38 3103dx3 i ith i yf 00¢ i] Ae 299 A (6uryopq pjob) aja, SJONSQIONION VAWV9-VAWV9 i 1 1 j L Osl ost 002 0ce Ove 092 082 00¢ O2¢e SLNNOD JONSGIONIOO FIGURE 25