Expressions for the energies of ground state yrast bands of even-even nuclei are obtained by angular-momentum projection on coherent phonon intrinsic states starting from a pairing plus quadrupole interaction. The results of the theory for a local minimum corresponding to zero deformation are investigated numerically and a good fit is found to the spectra of the xenon isotopes. Quadrupole moments and $B\left(E2\mathrm{}\right)\mathrm{}$'s are also discussed.

[NUCLEAR STRUCTURE Angular-momentum projection, even-even nuclei, yrast states, variational principle, xenon isotopes.]

The problem of two distinct pairs of identical particles is considered in Faddeev-Yakubovsky formalism. A set of seven coupled integral equations is obtained which is simplified by assuming separable $s$-wave forms for the interaction between any two particles. An angular momentum analysis is carried out to obtain the equations satisfied by the partial wave components.

[NUCLEAR STRUCTURE Faddeev-Yakubovsky formalism, four-body equations, distinct pairs of identical particles, separable two-body potentials, partial wave analysis.]

The nucleus ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ is studied with a wave function which represents a dineutron-plus-${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ cluster configuration. The result indicates that the dineutron clustering effect may be quite important for the structure of the three lowest states in this nucleus.

[NUCLEAR STRUCTURE ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$; calculated levels, $J$, $\pi$. Resonating-group method.]

Elastic scattering of 12-MeV ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ ions from targets made from Madagascar monazite shows no evidence for the presence of superheavy elements. Upper limits on fractional atomic abundances are 3 × ${10}^{-6\mathrm{}}$ for ekathorium and 6 × ${10}^{-8\mathrm{}}$ for eka-lead.

[NUCLEAR REACTIONS (${}_{}{}^7{}_{}{}^{}\mathrm{Li}$, ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$), $E=12.0\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$; target prepared from natural Madagascan monazite; deduced relative elemental concentrations; deduced upper limit for superheavy elements.]

The mean lifetime of the 3041-keV ($6_1^{+}$) state in ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ has been measured to be $\tau =4.5\mathrm{}\pm 0.3$ nsec. This value represents an improvement by a factor of 6 from previous measurements. States in ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ were populated by the ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}({}_{}{}^3{}_{}{}^{}\mathrm{He},n\gamma )$ reaction with $E_{3_{\mathrm{He}}}=10\mathrm{} \mathrm{and} 12$ MeV. The experimental value of $B\left(E2\mathrm{}\right)=27.6\mathrm{}\pm 1.8e^2$ ${\mathrm{fm}}^4$ for the ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ $6^{+}$ →$4^{+}$ transition together with the ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ca}$ $6^{+}$ →$4^{+}$ $B\left(E2\mathrm{}\right)\mathrm{}$ value provides a measure of the importance of the isovector giant quadrupole state on the effective $E2\mathrm{}$ operator which is used in shell-model calculations.

[NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}({}_{}{}^3{}_{}{}^{}\mathrm{He},n\gamma )$, $E=10,12\mathrm{}$ MeV; measured pulsed beam electronic timing; deduced $T_{\frac{1}{2}}$, $B\left(E2\mathrm{}\right)\mathrm{}$, effective charge. Natural target, Ge(Li) detectors.]

Employing the spin alignment obtained with inelastic $\alpha$ particle scattering at 0°, we have shown the $J^{\pi }$ of the 14.08 MeV level in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ to be $4^{+}$. The branching ratio of this level for $\alpha$ decay to the ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$ ground state is (9 ± 3)%; however, this ratio could be affected by interference with inelastic scattering to the $3\alpha$ continuum.

[NUCLEAR STRUCTURE ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ measured $\alpha$ decay angular correlation from 14.08 MeV level, deduce $J^{\pi }=4^{+}$.]

Examination of experimental data for $\gamma$-vibrational states reveals strong correlations between the nonadiabatic effect deviations of the $4_{\gamma }\to 2_{\gamma }$ to $3_{\gamma }\to 2_{\gamma }$ energy ratio from the adiabatic limit with $\Sigma i^{}\alpha _i^{}{}_{}{}^2$, where ${\alpha }_i$ are the coefficients for the $N=6,i_{\frac{13}{2}}$ neutron configuration in the expansion $|2_{\gamma }^{+}〉=\Sigma i^{}{\alpha }_i|2_q^{+}{}_p{}^{}{}_{}{}^{}〉_i^{}{}_{}{}^{}$. This correlation and other evidence presented indicate that in contrast to the usual exclusion of Coriolis effects in the structure of $\gamma$-vibrational states, Coriolis effects are the primary source of the nonadiabaticity.

[NUCLEAR STRUCTURE Nonadiabatic effects in $\gamma$-vibrational levels correlated to microscopic structure.]

The $g$ factor for the $J^{\pi }=6^{+}$ isomeric level in ${}_{}{}^{134}{}_{}{}^{}\mathrm{Te}$ can be reproduced within the experimental error by taking into account core polarization as well as velocity dependence corrections to the magnetic dipole operator. The wave function describing this isomeric state is obtained by diagnolizing a residual Gaussian interaction in a two-particle configuration space.

[NUCLEAR STRUCTURE ${}_{}{}^{134}{}_{}{}^{}\mathrm{Te}$; calculated levels, $J$, $\pi$, $g$ factor for isomeric $J^{\pi }=6^{+}$ level. Corrected $M1\mathrm{}$ operator: core polarization, velocity dependence.]

The renormalized random-phase approximation is treated for ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ using a density- and energy-dependent particle-hole force deduced from the nucleon-nucleon interaction. For the configuration space two main shells above and below the Fermi surface are used. The outcome is compared with the semiphenomenological treatment, which uses adjustable density-dependent $\delta$ forces or a surface-$\delta$ interaction, respectively.

[NUCLEAR STRUCTURE Renormalized random-phase approximation applied to ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ using microscopic particle-hole force.]

The structure of ${}_{}{}^{34}{}_{}{}^{}\mathrm{Cl}$ has been investigated with the reaction ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}({}_{}{}^3{}_{}{}^{}\mathrm{He},p)$. Angular distributions were obtained for 29 levels below 5 MeV excitation. Extracted $L$ values, combined with previous information, allow $J^{\pi }$ assignments of $3^{+}$ for states at 2181 and 2611 keV, negative parity for the $J=4\mathrm{}$ state at 3600 keV, $2^{-}$ for the level at 4514 keV, and $3^{-}$ ($2^{-}$) for states at 4416 and 4461 keV. Two new states are observed at excitation energies of 4693 and 4825 keV.

[NUCLEAR REACTIONS ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}({}_{}{}^3{}_{}{}^{}\mathrm{He},p),E\left({}_{}{}^3{}_{}{}^{}\mathrm{He}\right)=18\mathrm{}$ MeV; measured $\sigma (E_p,\theta )$; gas ${\mathrm{H}}_2$S target. ${}_{}{}^{34}{}_{}{}^{}\mathrm{Cl}$ deduced levels, $L$, $J$, $\pi$.]

The absolute and relative $\gamma$-decay strengths of the lowest $T=\frac{3}{2}$ levels in ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ are compared using the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(p,{\gamma }_0){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma ){}_{}{}^{13}{}_{}{}^{}\mathrm{C}$, and ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}({}_{}{}^3{}_{}{}^{}\mathrm{He},n\gamma ){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ reactions. By combining the present results with previous measurements, reduced asymmetries of $\frac{B\left({}_{}{}^{13}{}_{}{}^{}\mathrm{C}\right)}{B\left({}_{}{}^{13}{}_{}{}^{}\mathrm{N}\right)}-1=-0.07\mathrm{}\pm 0.13$, ${0.82}_{-0.6\mathrm{}}^{+1.2\mathrm{}}$, ≤0.83 ± 0.29, and -0.04 ± 0.14 are obtained for the ${\gamma }_0\mathrm{}\left(M1\mathrm{}\right)\mathrm{}$, ${\gamma }_0\mathrm{}\left(E2\mathrm{}\right)\mathrm{}$, ${\gamma }_1\mathrm{}\left(E1\mathrm{}\right)\mathrm{}$, and ${\gamma }_2\mathrm{}\left(M1\mathrm{}\right)\mathrm{}$ transitions, respectively. All of the known mirror $\gamma$ transitions in mass 13 are summarized and compared with theoretical calculations and with the analogous $\beta$ decays of ${}_{}{}^{13}{}_{}{}^{}\mathrm{B}$ and ${}_{}{}^{13}{}_{}{}^{}\mathrm{O}$. Upper limits of 2-7% are placed on the relative size of the isotensor transition matrix elements for the $M1\mathrm{}$ transitions. Changes in the radial wave functions induced by binding energy differences in ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ do not account for the observed asymmetry of the well known $E1\mathrm{}$ decays of the first excited states. This provides clear evidence of charge dependent parentage differences in the $T$-allowed components of the nuclear wave functions. For the lowest $T=\frac{3}{2}$ levels in ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ we find $\frac{{\Gamma }_{\gamma 0}}{\Gamma \left({}_{}{}^{13}{}_{}{}^{}\mathrm{C}\right)}=(0.396\mathrm{}\pm 0.030)\%$, $\frac{{\Gamma }_{\gamma 0}}{{\Gamma }_{p0\mathrm{}}\left({}_{}{}^{13}{}_{}{}^{}\mathrm{N}\right)}=(12.1\mathrm{}\pm 1.1)\%$, $\frac{{\Gamma }_{p0\mathrm{}}{\Gamma }_{\gamma 0}}{\Gamma \left({}_{}{}^{13}{}_{}{}^{}\mathrm{N}\right)}=(5.79\mathrm{}\pm 0.20)$ eV, ${\Gamma }_{\mathrm{total}}\left({}_{}{}^{13}{}_{}{}^{}\mathrm{C}\right)=(5.88\mathrm{}\pm 0.81)$ keV, and ${\Gamma }_{\mathrm{total}}\left({}_{}{}^{13}{}_{}{}^{}\mathrm{N}\right)=(0.86\mathrm{}\pm 0.12)$ keV. A new efficiency calibration standard at $E_{\gamma }=15.1\mathrm{}$ MeV is provided by our measurement of the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(p,{\gamma }_0){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ thick-target resonant yield, $Y_R=(6.83\mathrm{}\pm 0.22)\times {10}^{-9\mathrm{}} {\gamma }_0$'s per incident proton.

[NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(p,{\gamma }_0)$, $E=14.23\mathrm{}$ MeV resonance; ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}({}_{}{}^3{}_{}{}^{}\mathrm{He},n\gamma )$, particle-$\gamma$ coincidence; measured $\frac{{\Gamma }_{\gamma i}}{\Gamma }$ and deduced ${\Gamma }_{{\gamma }_i}$ and $\Gamma$ for ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ ($T=\frac{3}{2}$) and ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ ($T=\frac{3}{2}$); symmetry of mirror transitions.]

A concise formalism is developed which permits the efficient representation and generalization of several known techniques for deriving connected-kernel $N$-particle scattering integral equations. The methods of Kouri, Levin, and Tobocman and Bencze and Redish which lead to minimally coupled integral equations are of special interest. The introduction of channel coupling arrays is characterized in a general manner and the common base of this technique and that of the so-called channel coupling scheme is clarified. It is found that in the Bencze-Redish formalism a particular coupling array ahs a crucial function but one different from that of the arrays employed by Kouri, Levin, and Tobocman. The apparent dependence of the proof of the minimality of the Bencze-Redish integral equations upon the form of the inhomogeneous term in these equations is eliminated. This is achieved by an investigation of the full (nonminimal) Bencze-Redish kernel. It is shown that the second power of this operator is connected, a result which is needed for the full applicability of the Bencze-Redish formalism. This is used to establish the relationship between the existence of solutions to the homogeneous form of the minimal equations and eigenvalues of the full Bencze-Redish kernel.

[NUCLEAR REACTIONS Scattering theory, $N$-particle scattering, integral equations, channel coupling, array theory.]

The intent of this paper is to extend the scope of a previous calculation on the mass 13 nuclei. In that calculation, a structure model was presented which was successful in dealing with bound state properties of these nuclei. In the present calculation we use the same structure model, with no adjustable parameters, to calculate continuum properties of ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ through $\mathcal{L}$ operator methods. In particular, phase shifts and total elastic scattering cross sections for neutrons on a ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ target are calculated. It is found that agreement between theory and experiment is generally quite good.

[NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}\mathrm{}(n,n)\mathrm{}$, $E=0\mathrm{} \mathrm{to} 5$ MeV, calculated phase shifts and total cross sections; unified model of mass 13 system.]

Starting from the Brueckner-Hartree-Fock approximation and Reid's hard core nucleon-nucleon interaction, we calculate and parametrize the energy—and the density—dependence of the isoscalar, isovector, and Coulomb components of the complex optical-model potential in infinite nuclear matter, for energies up to 160 MeV. We then construct the optical-model potential in a finite nucleus. In a first step, we adopt a local density approximation which implies that the value of the complex potential at each point of the nucleus is the same as in a uniform medium with the local density. We compute the corresponding volume integrals per nucleon and mean square radii of the real and of the imaginary parts of the optical-model potential, in particular for protons scattered by ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$, ${}_{}{}^{120}{}_{}{}^{}\mathrm{Sn}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$. We compare these results with a compilation of empirical values and find that the calculated and experimental volume integrals are in good agreement but that the theoretical mean square radii are too small. We ascribe this discrepancy to the fact that our local density approximation does not include accurately the effect in a nonuniform medium of the range of the effective interaction. We include this range in a semiphenomenological way suggested by the Hartree approximation. With a reasonable value for this range parameter, which is the only one occurring in our work, good agreement is obtained between the theoretical and the empirical values of the volume integrals and mean square radii of the real and, to a lesser extent, of the imaginary parts of the optical-model potential, for mass numbers $12\le A\le 208$ and for energies $E$ up to 160 MeV. Our results are given in analytic form and can thus be used in analyses of experimental data. We also discuss the difference between the optical-model potentials for protons and for neutrons.

[NUCLEAR REACTIONS Calculation of the complex optical-model potential for finite nuclei from Reid's hard core interaction; comparison with a compilation of empirical potentials.]

The spectrum of protons from the ${}_{}{}^2{}_{}{}^{}\mathrm{H}\mathrm{}(n,p)2n$ reaction at $E_n=13.98\mathrm{}$ MeV has been measured at ${\theta }_p={1.6}^{\circ }$ with a magnetic-quadrupole-doublet charged-particle spectrometer. The experimental resolution of 200 keV is more than a factor of 2 improvement over that of previous measurements. The high energy peak in the proton spectrum due to the neutron-neutron final state interaction has been analyzed to yield a value of the neutron-neutron scattering length $a_{\mathrm{nn}}$. Comparison of these data with an impulse approximation calculation and with exact three-body calculations indicates a value of $a_{\mathrm{nn}}$ which is significantly more negative than the accepted value of $a_{\mathrm{nn}}=-16.6\mathrm{}\pm 0.5$ fm.

[NUCLEAR REACTIONS ${}_{}{}^2{}_{}{}^{}\mathrm{H}\mathrm{}(n,p)2n$, $E=13.98\mathrm{}$ MeV, ${\overline{\theta }}_p={1.6}^{\circ }$; measured $\sigma \left(E_p\right)$; deduced $n-n$ scattering length.]

The level structure of ${}_{}{}^{156}{}_{}{}^{}\mathrm{Dy}$ has been investigated with the ($p,t$) reaction at 29.9-MeV incident proton energy. Angular distributions were measured for transitions to 34 levels up to an excitation energy of 2.25 MeV in ${}_{}{}^{156}{}_{}{}^{}\mathrm{Dy}$. The ground-state $Q$ value for this reaction was determined to be $Q_0=-7.535\mathrm{}\pm 0.015$ MeV. Optical-model parameters were obtained self-consistently in the coupled-channels approach, and reaction calculations using these parameters demonstrated a very large destructive interference between direct and indirect transfer to the $2^{+}$ member of the ground-state band. The anomalously strong transition to the $J^{\pi }=0^{+}$ state at 674 keV is interpreted in the model of Van Rij and Kahana, leading to suggestions as to the structure of this state. The strong $L=2\mathrm{}$ transition to the $J^{\pi }=2^{+}$ state at 890 keV is also discussed. Further evidence is given on the structure of the $J^{\pi }=2^{+}$ state at 1520 keV, and the $J^{\pi }=4^{+}$ state at 1627.3 keV which has been suggested to belong to the "super" band in ${}_{}{}^{156}{}_{}{}^{}\mathrm{Dy}$.

NUCLEAR REACTIONS ${}_{}{}^{158}{}_{}{}^{}\mathrm{Dy}$($p,t$), $E_L=29.9\mathrm{}$ MeV, measured $\sigma \left(\theta \right)$, $Q_0$; CCBA and DWBA analysis; deduced $E_x$, $L$, $J^{\pi }$ of ${}_{}{}^{156}{}_{}{}^{}\mathrm{Dy}$ levels.

NUCLEAR STRUCTURE Model of Van Rij and Kahana applied to anomalous $0^{+}$ state; evidence for $\gamma$ vibration built on these states, and on nature of "super" band.

Angular distributions of the ($\alpha ,d$) reaction on ${}_{}{}^{32,33,34}{}_{}{}^{}\mathrm{S}$ and ${}_{}{}^{35,37}{}_{}{}^{}\mathrm{Cl}$ have been measured at 40 MeV bombarding energy. High-spin states of ${\mathrm{}\left(\mathrm{fp}\right)\mathrm{}}^2$ character in the final nuclei were identified on the basis of their strength and characteristic $L=4\mathrm{}$ and $L=6\mathrm{}$ angular distributions. For some states, spin assignments are suggested. The properties of the high-spin states are discussed.

NUCLEAR REACTIONS ${}_{}{}^{32,33,34}{}_{}{}^{}\mathrm{S}$, ${}_{}{}^{35,37}{}_{}{}^{}\mathrm{Cl}$ ($\alpha ,d$); $E=40\mathrm{}$ MeV; measured ($E_d,\theta$); enriched targets; deduced $L$ transfer.

Twelve ground state transitions up to about 5 MeV have been observed by the resonant scattering of bremsstrahlung from natural barium. Six of the first seven lines are from $2^{+}$ states in ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$ as evidenced by the angular distribution measurements and the correspondence of the energies with known levels in ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$. Presumably these arise from proton configurations. Of the remaining five lines four have distributions expected for spin 1 levels. The linear polarization of the scattered radiation for the first of these, at 4027 keV, suggests a negative parity. It is likely that this state is derived from the superposition of the one phonon quadrupole and octupole transitions in ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$. Comparison with the results of other studies indicates that the other three lines are from neutron particle-hole states in ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$. Widths in terms of the ground state branching ratios and the partial widths for decays to the ground state are given for the identified levels.

NUCLEAR REACTIONS ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$ ($\gamma ,\gamma$), ($\gamma ,{\gamma }^{\prime }$); bremsstrahlung $E=1.5-5.1\mathrm{}$ MeV; measured $\sigma$ (96°) and $\sigma$ (126°), $\mathrm{LP}$; deduced $J$, $\pi$, $\Gamma$. Natural targets.

The most intense $\gamma$ rays from capture of thermal neutrons in ${}_{}{}^{86}{}_{}{}^{}\mathrm{Kr}$ are measured with pair and Compton-suppressed spectrometers. The $Q$ value for the reaction is found to be 5515.4 ± 0.8 keV and the reaction cross section is estimated to be about 3 mb. Energy levels of the ${}_{}{}^{87}{}_{}{}^{}\mathrm{Kr}$ nucleus observed in this study are compared with those seen in reaction studies and in the $\beta$ decay of ${}_{}{}^{87}{}_{}{}^{}\mathrm{Br}$.

NUCLEAR REACTIONS ${}_{}{}^{86}{}_{}{}^{}\mathrm{Kr}$ $n_{\mathrm{the}},\gamma$; measured $E_{\gamma }$, $I_{\gamma }$; Ge(Li) Compton suppression and pair spectrometers; deduced $Q$, $\sigma$; ${}_{}{}^{87}{}_{}{}^{}\mathrm{Kr}$ deduced levels, $J$, $\pi$.

Using bremsstrahlung produced with 6.6 and 9.7 MeV beams, nuclear resonance fluorescence measurements were made on targets of ${}_{}{}^{206,207,208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$. Ground state transition widths for previously unknown energy levels with widths $\gtrsim 1$ eV were obtained. An interpretation of several of these levels in terms of a particle-core weak coupling model is suggested.

NUCLEAR REACTIONS ${}_{}{}^{206,207,208}{}_{}{}^{}\mathrm{Pb}$, ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$$\gamma ,\gamma$; resonance fluorescence with 9.7 and 6.6 MeV bremsstrahlung. Levels measured $E_{\gamma }$, $I_{\gamma }$ at 125° (lab); deduced $\frac{g\Gamma _0^{}{}_{}{}^2}{\Gamma }$.