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Energy levels and transition probabilities for deformed nuclei
Abstract
Higher-order energy corrections and intensity rules for K-allowed and K-forbidden transitions are discussed on the basis of a generalized Peierls-Yoccoz theory, which includes K mixing. The results put a number of predictions of the rotational model on a firmer basis, but show in addition some marked differences. A comparison with experimental data is given.
... The AMP enables to calculate rotational energies straightforwardly. The J(J + 1) rule of the excitation energy with the moment-of-inertia is derived from the AMP under a reasonable approximation for well-deformed heavy nuclei [3,7,8,[13][14][15]. However, for light or weakly-deformed nuclei, it ...
... Additionally, we present a general formulation for the rotational energy. The derived formula is a generalization of the previous ones in Ref. [3,7,8,[13][14][15], while the additional terms could be important in the rotational energy and the moment-ofinertia for light or weakly-deformed nuclei. ...
... The J(J +1) rule of rotational energy and the moment-of-inertia of nuclei connected with Eq. (3) were discussed in Refs. [3,7,8,[13][14][15]. If the overlap function Φ 0 |e −iĴyβ |Φ 0 has a sharp peak at β ≈ 0, the energy spectrum is close to the J(J + 1) rule. ...
The origin of the rotational band of nuclei has been investigated by the angular momentum projection (AMP) on the axial Hartree-Fock solutions, by using the semi-realistic effective Hamiltonian M3Y-P6. The rotational energy is decomposed into contributions of the individual terms of the Hamiltonian, and their ratios to the total rotational energy are calculated. Except for light or weakly-deformed nuclei, the ratios of the individual terms of the Hamiltonian are insensitive to nuclides and deformation. The contributions of kinetic energies are large and close to the rigid-rotor values, although those of central forces are sizable. For light or weakly-deformed nuclei, the ratios significantly depend on nuclei and deformation. The contributions of noncentral forces are not negligible. Regardless of nuclides, the attractive forces decrease the moment-of-inertia, and the repulsive forces increase it. A general formula for the rotational energy is derived on the basis of the AMP, which suggests that higher-order terms of the cumulant expansion play roles in the rotational energy and the moment-of-inertia for light or weakly-deformed nuclei.
... Λ 2n and Λ 2n+2 /Λ 2n in equation(8) for the HF+BCS results of12 34 Mg and40 80,100 Zr. The symbols correspond with the n values indicated in the inset. ...
By applying the angular-momentum projection to the self-consistent axial mean-field solutions with the semi-realistic effective Hamiltonian M3Y-P6, the pairing effects on the pure rotational energy of nuclei, i.e. the rotational energy at a fixed intrinsic state, have been investigated. While it was shown at the Hartree–Fock (HF) level that the individual terms of the Hamiltonian contribute to the rotational energy with ratios insensitive to nuclides except for light or weakly-deformed nuclei, the pair correlations significantly change the contributions, even for the well-deformed heavy nuclei. The contribution of the interaction to the rotational energy is found to correlate well with the degree of proximity between nucleons, which is measured via the expectation value that two nucleons exist at the same position. While the nucleons slightly spread as the angular momentum increases at the HF level, accounting for the positive (negative) contribution of the attractive (repulsive) components of the interaction, the pair correlations reduce or invert the effect.
The Peierls-Yoccoz (PY) rotational energy of nuclei has been analyzed by the angular-momentum projection on the axial Hartree-Fock solutions, by using the semi-realistic effective Hamiltonian M3Y-P6. The rotational energy is decomposed into contributions of the individual terms of the Hamiltonian, and their ratios to the total PY rotational energy are calculated. Except for light or weakly deformed nuclei, the ratios of the individual terms of the Hamiltonian are insensitive to nuclides and deformation. The contributions of kinetic energies are large and close to the rigid-rotor values, although those of central forces are sizable. For light or weakly deformed nuclei, the ratios significantly depend on nuclei and deformation. The contributions of noncentral forces are not negligible. Regardless of nuclides, the attractive forces decrease the moment of inertia, and the repulsive forces increase it. A general formula for the PY rotational energy is derived, which suggests that higher-order terms of the cumulant expansion play roles in the rotational energy and the moment of inertia for light or weakly deformed nuclei.
In the first part of these lectures (hereafter referred to as I) we introduced the mathematical framework of the Generator Coordinate Method (GCM) and outlined how nuclear positions can be used as generator coordinates to construct a non-adiabatic theory of molecules (Van Leuven and Lathouwers, this volume).
All available experimentally determined absolute gamma-ray transition probabilities between different intrinsic states of deformed odd-mass nuclei in the rare earth region 153 ≦ A ≦ 181 and in the actinide region 227 ≦ A ≦ 251 are compared with theoretical transition probabilities (Weisskopf and Nilsson estimate). Systematic deviations from the theoretical values are found. Possible explanations for these deviations are given. The discussion includes Coriolis coupling, ΔK = ±2 band-mixing effects and pairing interaction.
Possible deviations from the Alaga rules based on the use of projection-integral wave functions are investigated. It is shown that while such deviations, in principle, exist, they identically vanish in the region of heavy deformed nuclei, if it is assumed that these nuclei can be described by "semirigid" intrinsic wave functions. [NUCLEAR MOMENTS Deformed nuclei; calculated B(E2) deviations. Projection integral wave functions.]
It is shown that a rigorous application of the Peierls-Yoccoz angular momentum projection method leads, in the case ofK=0 rotational bands, to a simple exact expression for the projected energyE
I
, which may be especially suitable for numerical calculations. On the basis of this energy law and without making any assumption for the overlapsn(Β) andh(Β), a finite expansion ofE
I
in powers ofI(I
+1) is obtained and discussed. Using this finite series a microscopical variable moment of inertia model is presented.
Band-mixing and decoupling factors are obtained for the coupling of an odd particle to a deformed core. The derivation avoids the use of commutation rules in a rotating frame and is easily generalized to more microscopic models of deformed nuclei.
The concept of an ideal collective coordinate is introduced by means of the following example: Consider a one-dimensional vibration of a many-body system in the sense that a large subset of states |n> of the system exhibits an energy spectrum and relative transition probabilities following the laws of the (in general anharmonic) oscillator described by H (palpha, alpha) (alpha|n)=omegan(alpha|n). We suppose the set of many-body states |n> to extend indefinitely, and we take the transform |alpha>=|n> (n|alpha) to define a many-body generating state of the band which is precisely localized in alpha space. The basic assumption of collectivity, that changing the state of at most a few particles cannot much alter the value of alpha, is shown to be sufficient to derive a phenomenological theory from the many-body starting point. The phenomenological aspects of a recent theory of rotations due to Villars is seen to be contained in the above formulation as a special case. A brief review is given of the generator coordinate and similar projection methods in order to exhibit their relationship with the present method.
Deformed charge distributions of 20Ne and 28Si are determined from electron scattering within a framework of the rotational model. Deformed Fermi distributions are assumed
for the intrinsic charge distributions of these nuclei and the DWBA analysis is carried out for the ground-band transitions.
The results are compared with (p,p') and (α,α') results. The validity of the scattering analysis using the rotational model is also examined to some extent.
For this purpose, previous rotational model results for 20Ne, 24Mg and 28Si with Nilsson-like intrinsic states are compared with the results using angular momentum projection.
The generator coordinate method is formulated so as to be suitable for description of the nuclei in the transition region.
To take into account the pairing correlation, the BCS wave function constructed with various single particle states in a deformed
potential well is used as an intrinsic wave function, from which eigenstates of angular momentum are projected out. Using
the quadrupole deformation parameters, β and γ, and the gap parameters, Δn and Δp, as generator coordinates, the trial wave function is generated by superposing the projected wave functions with various
of the generator coordinates. Such a choice of the generator coordinates enables us to treat the rotation, the surface vibration
and the pairing vibration in a unified way. The generator function, which serves as a weight function, is obtained by solving
an integral equation derived through the variational principle. It is shown that our projected wave functions have the same
symmetry properties as those of wave functions given by the Bohr model.
The theory of Rouhaninejad and Yoccoz concerned with the self-consistent calculation with projection method is extended to take into account the pairing correlation. The variational method is used in which the trial function involving a definite angular momentum is generated by using the BCS wave function. The first order correction to the self-consistent field due to the rotation contains two parts, with K=0 and with K=1. The correction to the density matrix with K=1 and that to the pairing tensor with K=1 is proportional to sqrt{I(I+1)}. They do not, however, contribute to the rotational energy in the lowest order of the inverse of the expectation value of the square of the angular momentum.
Vibrations of deformed even nuclei are treated as vibrations in the intrinsic system projected out into rotational bands in the laboratory. A vibrational quantum is viewed as a quantized surface ripple, with a good K but not J, in analogy to a nucleon in the Nilsson well. The intrinsic Hamiltonian is taken to be of the Bohr-Mottelson type, with the parameters Blm, Ctm and β0 defining the deformed field. The intrinsic problem is solved in terms of the phonons of the spherical field, first in perturbation theory and then exactly. The Bogoyubov-Valatin transformation for bosons is used in the latter. By the Peierls-Yoccoz projection method, the physical states are found as certain superpositions of spherical-phonon states with various phonon numbers. The present model reproduces the usual results of the Bohr-Mottelson model; in addition, it suggest a K = 1 quadrupole-vibrational band. Finally, the coherence, or collectivity, of the projected states is discussed in terms of the B(E2) value between the two lowest states. The expected qualitative behaviour is established. The B(E2) value increases as deformation sets in; slow convergence encumbers the study of large deformations. Altogether, the present work continues work continue the phonon descriptiopn of spherical nuclei to the permanently deformed ones; it forms a rather detailed examination of the long-alleged equivalence of deformation and vibration.
The ground-state rotational bands of deformed even nuclei are described by the two-parameter formula E = a[sqrt(1 + bJ(J + 1)) - 1]; expanded, it gives the form of the usual power series in J(J + 1). The formula is deduced from experimental level systematic; alternatively, it can be derived from nuclear hydrodynamics. With the two parameters determined from two medium-high levels, our formula describes rather well the other levels, whereas the usual power series tends to diverge for high J-values. Other recent descriptions of rotational spectra are appreciably more complicated. Department of Physics, Siltavuorenpenger 20C, Helsinki 17, Finland.
A method for projecting spurious rotational states from many-particle nuclear wavefunctions is extended to wavefunctions generated by a deformed shell model of axial symmetry. For completeness, the projection procedure is reviewed [19]. According to this method the deformed shell model (SM) Hamiltonian is altered by a correction term such that the new Hamiltonian is translation- and rotation-invariant, i.e., it depends only on intrinsic coordinates when it is transformed to a body-fixed system. The eigenfunction of which are obtained by diagonalizing this Hamiltonian in the shell model basis separate into an intrinsic function and a collective factor depending on the center of mass coordinates and Euler angles. By identifying the spurious rotational states this set of functions can be made nonredundant so that the functions are a complete substitute for a set of realistic intrinsic functions. For shell models with axial symmetry we introduce only two Euler angles. It turns out that the intrinsic function is characterized by a good projection of the total angular momentum on the symmetry axis of the nucleus. Applications for the Ωπ = 0+, T = 0 bandheads of 20Ne are given and good agreement with the experimental spectrum is obtained. A study of the potential energy surface leads to the conclusion that it contains spurious rotational energies.
The basis for much of our present understanding of nuclear structure derives from the study of mean field approximations (such as Hartree-Fock or Bardeen-Cooper-Schrieffer) and of small deviations from the mean field (random phase approximation and the cranking model). This review is devoted to the study of a theoretical framework (and of some applications of it) which provides not only a sound derivation of all these approximations but a basis for going beyond them. The approach may be characterized broadly as the application of Heisenberg matrix mechanics to the nuclear many-body problem. It utilizes a calculus for matrix elements of suitably chosen simple operators between exact eigenstates of the Hamiltonian. In the first class of investigations, in which single fermion operators were chosen as the suitable operators, one ends with a theory providing a justification for and generalization of various core-particle coupling models and a basis for nuclear field theory. In a further group of studies of matrix elements of multipole and/or pair operators, the collective behavior of even nuclei can be examined, divorced from their coupling to neighboring odd nuclei. Various investigations carried out over two decades are described, including theoretical foundations, applications to schematic models, and applications to vibrational and rotational problems. A common theoretical thread is that the calculations are done uniformly in fermion shell model space in a representation in which the Hamiltonian is diagonal. In the concluding remarks, we point to new developments which may alter these considerations profoundly by enlarging the framework in which they may be carried out.
We review the merits and shortcomings of the Born-Oppenheimer separation and suggest a nonadiabatic approach to molecular spectra using the generator-coordinate method. The adiabatic approximation and the new scheme are worked out in parallel for diatomic molecules.
The existence of rotational bands in nuclei of the 2s-1d shell (16 < A < 40) and to a lesser extent in those of the 1p shell (4 < A < 16) has been recognized for a long time. The main features of such bands are the approximate proportionality to J(J + 1) of the energy levels and the strong quadrupole transition moments and static moments. Although these aspects are not as striking in a light nucleus as they are among the heavy deformed ones, the interpretation of the experimental data in terms of a rotating deformed intrinsic state seems unescapable. This article is concerned with the calculation and the study of the deformations in light nuclei with the Hartree-Fock (H.F.) method.
For even-even nuclei, the excitation energy E2 and the reduced transition probability B(E2) between the ground state and the first excited 2+ state have been considered. On the basis of different models, it is shown that for a nucleus [N, Z] the relations E2[N, Z]+E2[N+2, Z+2]-E2[N+2, Z]-E2[N, Z+2]≈0 and B(E2)[N, Z]+B(E2)[N+2, Z+2]-B(E2)[N+2, Z]-B(E2)[N, Z+2]≈0 hold good, except in certain specified regions. The validity of these difference equations is tested with the available experimental data. The difference equation of Ross and Bhaduri is shown to follow from our approach. Some predictions of unmeasured E2 and B(E2) values have been made.NUCLEAR STRUCTURE Simple relations for E2 transition probabilities and excitation energies of first excited 2+ states in even-even nuclei.
The generator-coordinate method is applied to the rotational spectra of the ground state of even nuclei. The trial functions are generated from the Nilsson well, taking into account pairing correlations for fixed values of the deformation parameter δ. This wave function is projected onto eigenstates of angular momentum and is used to evaluate the expectation value of the Hamiltonian, which consists of the kinetic energy and the interactions between nucleons. A Gaussian form with Rosenfeld mixture is assumed for the nucleon-nucleon potential. Numerical calculations are performed for isotopes of Hf. Without changing the values of parameters, characteristic features of the energy spectra are reproduced and agreement with experiment is good. The quadrupole moments of various rotational states, as well as E2 transition matrix elements, are also calculated. To make more extensive comparisons, the kernels used in the above calculations are expressed in terms of parameters, which are selected to give agreement with experimental spectra up to the second excited states for each isotope. Then energy spectra are calculated using these adjusted parameters for all rotational nuclei with mass number greater than 100. Good agreement with experimental data is obtained.
In this review it is shown how the physical understanding of clustering aspects allows the formulation of a unified microscopic theory of nuclear structure and nuclear reactions. In particular, the fundamental importance of the Pauli principle is stressed in resolving the contradictions between different collective and single-particle aspects of the atomic nuclei. After the general formulation of the method, which can be applied also to other many-particle systems, several illustrative examples are given for quantitative studies of bound-state scattering and reaction problems, for semi-quantitative studies and for the derivation of general nuclear properties.
For the class of nuclei which are 'strongly deformed' it is possible to introduce the idea of an empirically measurable static nuclear shape. The limitations of this concept as applied to nuclei (fundamentally quantum-mechanical objects) are discussed. These are basically the limitations of the rotational model which must be introduced in order to define and measure nuclear shape. A unified discussion of the ways in which the shape has been parametrized is given with emphasis on the fact that different parametrizations correspond to different nuclear structures. Accounts of the various theoretical procedures for calculating nuclear shapes and of the interaction between nuclear shapes and nuclear spectroscopy are given. A coherent account of a large subset of nuclei (strongly deformed nuclei) can be given by means of a model in which the concept of nuclear shape plays a central role.
The angular momentum projection technique is worked out for systems with an intrinsic asymmetric deformation. These include asymmetric top molecules and triaxial nuclei. A parameterisation of the rotation group is introduced which allows an approximate but analytical evaluation of angular momentum projected matrix elements. Via symmetric orthonormalisation a quantal definition of the moments of inertia is obtained.
For pt.I see ibid., vol.15, no.9, p.2785-79 (1982). The implications of intrinsic symmetries on the structure of rotational secular equations in the angular momentum projection formalism are studied for both the exact and approximate versions. The symmetry group of the rigid rotor, D2, is used to illustrate formal results.
The generator coordinate method is presented as a nonadiabatic theory of molecular dynamics. Details are worked out for diatomics.
The escape width originating from a collective rotational state as a doorway state is calculated in the framework of the shell model theory of nuclear reactions. The rotational states are constructed by Yoccoz-Peierls angular momentum projection from deformed intrinsic states which are described by BCS wave functions. For some rare earth nuclei it is shown that there result escape widths of the order of magnitude of 20–100 keV.
Generator-coordinate methods make use of variational parameters in microsopic or many-body wave functions to describe collective degrees of freedom. The use of these methods in symmetry restoration, nuclear rotational and vibrational states is described. Numerical applications in various nuclear structure problems are briefly reviewed. The structure of the integral wave equation appearing in the method is briefly discussed. Applications of these methods to study nuclear reactions are reported. These methods permit microscopic descriptions of collective nuclear properties, especially in low-energy situations requiring the use of antisymmetrized wave functions.
By using generator coordinates as state labels we formulate the generator-coordinate theory of collective motion as non-orthogonal representations of the many-body Schrödinger equation in a subspace of the Hilbert space of many-body state vectors. The weight function of the usual generator-coordinate theory is generalized to become the components of the state vector in one of the two bi-orthogonal representations labelled by generator coordinates. The well-known case of the Gaussian-overlap approximation is studied in order to show how the new formalism also permits a solution of the problem using the original real generator coordinates. The concept of generator coordinates is clarified by studying (i) the connection between the redundancy of generator coordinates and the linear dependence of base vectors labelled by these generator coordinates, and (ii) the construction of states having the required properties under translation and rotation. Finally, the consideration of rotational properties leads to a double-projection method for constructing internal states of good angular momentum and for removing spurious states of c.m. motion.
A basis of two particle intrinsic states for 22Ne and 26Mg is defined in terms of the Hartree-Fock solutions for 20Ne and 24Mg. Mixing between the different intrinsic states is estimated from a deformed shell model. The model proceeds to describe the lowest 12 positive parity states in 22Ne and the lowest 15 in 26Mg. Spin, parity and K-quantum number assignments are proposed. These results are also compared with the existing shell model and projection model interpretations of 22Ne and 26Mg.
The variation-after-projection Hartree-Fock method is described. Angular momentum projection algorithms which are practical to use in small and large bases, for axially symmetric as well as asymmetric states, for doubly even, doubly odd as well as odd-even nuclei, are discussed in detail. The method is successfully used to calculate the ground band structures of 20Ne (axial) and 24Mg (non-axial), in a five-major-shell basis with the Saunier-Pearson semi-realistic N-N interaction. The Operators J2, Q20 and Rs2 are used as cranking constraints. Level schemes, static moments, E2 transition strengths, charge radii, elastic and inelastic electron scattering form factors are computed for the two nuclei studied. Comparison are made with experimental data, with results obtained in other microscopic calculations, and with predictions in the rotational model.
The character of the rotational motion implied by the highly successful adiabatic rotational model is examined with the objective of determining the ground rules for a microscopic rotational theory based on this model. It is deduced that the adiabatic model implies rotational flow, i.e. rigid-body flow except that the particles are not frozen in position. However, the rotational flow of the adiabatic model does not exclude the possible existence in the nucleus of rotationally invariant clusters. As a particular example of such clustering, we propose a twofluid model of nuclear rotations to describe the possibility of a centrally located rotationally invariant superfluid core. It is shown that the inertia tensor for rotational flow is the rigid-body tensor. But it is also shown that clusters would participate in the rotational flow as if they were elementary particles with their masses concentrated at their centers of mass and would thereby effect a reduction of the moments of inertia. Thus the fact that observed moments of inertia, for nuclei described by the adiabatic rotational model, are considerably smaller than the rigidbody moments evaluated for the nucleon mass distribution can be attributed to the presence of cluster correlations. The Inglis cranking model is examined in the light of this interpretation.
Within the framework of the projection theory of collective motion, a microscopic description of the rotational energy with band-mixing is formulated using a method based on an inverse power perturbation expansion in a quantity related to the expectation value of the operator Jy2. The reliability of the present formulation is discussed in relation to the difference between the individual wave functions obtained from the variational equations which are established before and after projection. In addition to the various familiar quantities which appear in the phenomenological energy formula, such as the moment of inertia parameter, the decoupling factor and the band-mixing matrix element for |ΔK| = 1, other unfamiliar quantities having the factors with peculiar phases, (- 1)J+1 J(J + 1), (- 1)J+3/2(J - 1/ 2) (J + 1/2) (J + 3/ 2), (- 1)J+1/2(J + 1/ 2) J(J + 1), (- 1)J J(J + 1) (J - 1) (J + 2) and [J(J + 1)]2 are obtained. The band-mixing term for |ΔK| = 2 is also new. All these quantities are expressed in terms of two-body interactions and expectation values of the operator Jym, where m is an integer, within the framework of particle-hole formalism. The difference between the moment of inertia of an even-even and a neighbouring even-odd nucleus, as well as the effect of band-mixing on the moment of inertia are studied. All results are put into the forms so as to facilitate comparisons with the corresponding phenomenological terms and also for further application.
The Coulomb excitation of Tb159, Ho165 and Tm169 has been studied using principally 60 MeV O16 ions as projectiles. Both gamma ray and conversion electron spectra were taken. Rotational bands observed at 514 and 687 keV in Ho165, at 580 (and possibly ≈ 1280) keV in Tb159 and at 570 (and possibly ≈ 1170) keV in Tm169 have been assigned as collective (gamma vibrational) bands. The bands at 361 keV in Ho165, at 348 and 971 keV in Tb159 and at ≈ 900 keV in Tm169 are ascribed to intrinsic (Nilsson) configurations. The properties of these bands are discussed in some detail.By multiple excitation 6 or 8 members of each ground state rotational band were excited. The analysis of the energies of these levels indicates the presence of higher-order Coriolis terms in the rotational energy formula in the cases Tb159 and Tm169.
The 1042-kev state in Cm ²â´â´, which is populated in the BETA ; decay of 10.1-hv Am²â´â´, has previously been assigned (K,I pi ) = (6,6/sup +/; ). This assignment has now been corfirmed through an angular correlation ; measurement, and it was found that the 746-kev gamma ray is a mixture of 48% ; quadrupole (E2) and 54% dipole (M1). The half life of the 1042-kev state was ; determined by measurenrent of delayed coincidences between BETA particles and ; conversion electrons. The result is 34 plus or minus 2 ms. Comparison with ; the single particle estimate shows that the E2 transitions from the delayed state ; are hindered by factors of about 10¹°. This reflects the fact that the ; transitions are four times forbidden in K. The relative transition probabilities ; for the E2's from the delayed state are shown to agree well with a recent ; theoretical estimate. (auth);
A displacement of energy levels (so-called odd-even shift) was observed ; in the K = 0 rotation band of odd deformed nuclei. It is shown that the theory ; of collective rotation proposed by Peierls and Yoccoz explains this displacement, ; but predicts in addition different moments of inertia for the even and odd spin ; levels. A discussion is presented in terms of the unified model, on the basis of ; which an explanation of the odd-even shift was given previously. (auth);
It is shown that the Peierls-Yoccoz theory of nuclear collective motion ; gives a natural explanation for the decoupling effect in the case of K = 1/2 ; rotation bands of oddmass nuclei and for the odd-even shift in the case of K = 0 ; rotation bands of odd nuclei. Furthermore, the theory implies some additional ; modifications of the I(I + 1) energy law. An estimate of the effects is given. ; (auth);
The β- decay of 13 y 154Eu (3-) to the 123 keV 2+ level and the 371 keV 4+ level in 154Gd has been investigated in a β-γ coincidence experiment by means of a six-gap beta-ray spectrometer. We found the intensities (9.2±1.5)% and (0.19±0.05)% for the two beta groups. Only for the strong group was it possible to study the shape; the experiment shows a pronounced deviation from the statistical shape corresponding to the value (1.0±0.3) for the parameter Y2 of the modified Bij approximation. This value is also consistent with the available β-γ angular correlation data. The branching ratios to the 2+ and 4+ levels for the components of multipole orders 1 and 2 both strongly violate the generalized intensity rules for K-forbidden transitions in deformed nuclei, and a comparison is made with similar cases.
The gamma-rays of Tm-169 excited through electron capture in Yb-169 have been studied by use of a bent-crystal gamma-ray spectrometer. A source of high specific activity was produced from highly enriched Yb-168 by neutron capture. In addition to gamma rays previously reported we observe gamma rays with energies of 117.25 keV, 156.66 keV and 336.5 keV. The 117.25 keV gamma ray probably is due to the depopulation of a state at 433.44 keV. The relative intensities of most of the gamma rays were measured to ±5%. Electron capture branching ratios to the various levels were derived. Branching ratios for transitions between different rotational bands were studied in detail. Most M1 and E2 interband branching ratios are in agreement with predictions based on recent formulas by Bohr and Mottelson while all E1 branching ratios differ from these predictions by an order of magnitude.