Motivated by the recent paper arXiv:2602.11815 [nucl-th], we calculate in these notes the spectral functions (i.e. imaginary parts) of the NN-potentials as they arise from $2\pi$-exchange with single and double Roper resonance excitation. In contrast to the full one-loop calculation, the spectral functions of the isoscalar and isovector central and tensor potentials can be given in simple analytical form. The pertinent momentum-space potentials $V_C(q), W_C(q), V_T(q), W_T(q)$ are obtained via subtracted dispersion relations. This representation allows also to include a regulator function, that tames high-momentum components of the chiral $2\pi$-exchange. The calculation is extended to $2\pi$-exchange with combined $\Delta(1232)$-isobar and Roper resonance excitation.
We discuss how to generate entangled Bell states of two nucleons using unpolarized nucleon-nucleon scattering or the exclusive deuteron breakup reaction. We follow the the approach of Z. X. Shen et al., arXiv:2510.24325v1 [nucl-th], where Bell states were identified in unpolarized proton-proton elastic scattering. We confirm these results and show that, in the unpolarized proton-deuteron breakup reaction, it is also possible to generate proton-proton entangled Bell states in kinematically complete proton-proton quasi-free scattering (QFS) and final-state interaction (FSI) configurations. We also discuss an experimental setup that, by exploiting such entangled states, could enable the teleportation of quantum mechanical states in a three-proton system. Such an experiment requires triple coincidences among the outgoing nucleons, which precludes the use of entangled Bell states generated with extremely polarized incoming particles. Since counting rates for unpolarized reactions are much higher than for polarized ones, the present results open a pathway toward searching for signatures of quantum state teleportation in hadronic systems.
Kjeld Beeks, Georgy A. Kazakov, Fabian Schaden
et al.
State-resolved laser spectroscopy at the 10$^{-12}$ precision level recently reported in $arXiv$:2406.18719 determined the fractional change in nuclear quadrupole moment between the ground and isomeric state of $^{229}\rm{Th}$, $ΔQ_0/Q_0$=1.791(2) %. Assuming a prolate spheroid nucleus, this allows to quantify the sensitivity of the nuclear transition frequency to variations of the fine-structure constant $α$ to $K=5900(2300)$, with the uncertainty dominated by the experimentally measured charge radius difference $Δ\langle r^2 \rangle$ between the ground and isomeric state. This result indicates a three orders of magnitude enhancement over atomic clock schemes based on electron shell transitions. We find that $ΔQ_0$ is highly sensitive to tiny changes in the nuclear volume, thus the constant volume approximation cannot be used to accurately relate changes in $\langle r^2 \rangle$ and $Q_0$. The difference between the experimental and estimated values in $ΔQ_0/Q_0$ raises a further question on the octupole contribution to the alpha-sensitivity.
Neutron emission spectra (NES) of $^{232}$Th+n interaction provide strong evidence of angular anisotropy of secondary neutron emission, another evidence might be predicted in $^{232}$Th%(n,F)% prompt fission neutron spectra (PFNS). In case of NES observed angular anisotropy is presumably due to angular dependence of elastic scattering, direct excitation of collective levels and preequilibrium emission of (n,nX) neutrons. In $^{232}$Th+n direct excitation data analysis, ground state band levels are coupled within rigid rotator model, while those of beta bands, gamma bands and octupole band theta are coupled within soft deformable rotator model. NES of $^{232}$Th+n at En of 6, 12, 14, 18 MeV exhaustively described. The net effect of these procedures for En up to 20 MeV is the adequate approximation of angular distributions of $^{232}$Th$(n,nX)$ first neutron inelastic scattering in continuum, which corresponds to U of 1.2-6 MeV excitations of $^{232}$Th. The contribution of $^{232}$Th$(n,F)$ PFNS to the NES is exceptionally low. PFNS anisotropy occurs because some portion of $(n,nX)$ neutrons might be involved in exclusive prefission neutron spectra. In $^{232}$Th$(n,xnf)$ reactions PFNS demonstrate different response to forward and backward (n,xnf) neutron emission relative to the incident neutron momentum, when compared with $^{235}$U$(n,xnf)$ or 239Pu(n,xnf) reactions. Average energy of (n,xnf) neutrons depends on the neutron emission angle theta, i.e. fission cross section, prompt neutron number and total kinetic energy are shown to vary with the angle theta as well. Exclusive neutron spectra ($n,xnf)$ at theta equal to 90 degrees are consistent with observed $^{232}$Th$(n,F)$ and $^{232}$Th$(n,xn)$ reaction cross sections within En from 1 to 20 MeV energy range.
The nuclear scientific community views $^{232}$Th as an option for fuel in the future nuclear energy program. Numerous experimental studies have been conducted to determine the cross-section; however, very few have been performed to calculate the total neutron multiplicity above 10 MeV energy. In this work, we have compared the experimental data of average neutron multiplicity at different incident energies from EXFOR with the evaluated data from ENDF/B-VI, JENDL-4.0, and the calculated data from TALYS-1.96. The experimental data are in good agreement with the evaluated data from both the ENDF/B-VI and JENDL-4.0 libraries and at high incident energy (7 MeV), the TALYS-1.96 data are also in agreement with the experimental data.
Population of the 8 eV $^{229m}$Th isomer via the second nuclear excited state at 29.19 keV by means of coherent x-ray pulses is investigated theoretically. We focus on two nuclear coherent population transfer schemes using partially overlapping x-ray pulses known from quantum optics: stimulated Raman adiabatic passage (STIRAP), and successive $π$ pulses. Numerical results are presented for three possible experimental setups. Our results identify the Gamma Factory as the most promising scenario, where two ultraviolet pulses combined with relativistically accelerated ions deliver the required intensities for efficient isomer population. Our simulations require knowledge of the in-band and cross-band nuclear transition probabilities. We give theoretically predicted values for the latter and discuss them in the context of recent experiments.
The physical conditions for the emergence of the extremely low-lying nuclear isomer $^{229m}$Th at approximately 8 eV are investigated in the framework of our recently proposed nuclear structure model. Our theoretical approach explains the $^{229m}$Th-isomer phenomenon as the result of a very fine interplay between collective quadrupole-octupole and single-particle dynamics in the nucleus. We find that the isomeric state can only appear in a rather limited model space of quadrupole-octupole deformations in the single-particle potential, with the octupole deformation being of a crucial importance for its formation. Within this deformation space the model-described quantities exhibit a rather smooth behaviour close to the line of isomer-ground state quasi-degeneracy determined by the crossing of the corresponding single-particle orbitals. Our comprehensive analysis confirms the previous model predictions for reduced transition probabilities and the isomer magnetic moment, while showing a possibility for limited variation in the ground-state magnetic moment theoretical value. These findings prove the reliability of the model and suggest that the same dynamical mechanism could manifest in other actinide nuclei giving a general prescription for the search and exploration of similar isomer phenomena.
A. Kovalík, A. Kh. Inoyatov, L. L. Perevoshchikov
et al.
The 9.2 keV nuclear transition in $^{227}$Th populated in the $β$-decay of $^{227}$Ac was studied by means of the internal conversion electron spectroscopy. Its multipolarity was proved to be of mixed character M1+E2 and the spectroscopic admixture parameter $δ^2$(E2/M1)=0.695 $\pm$ 0.248 (|$δ$(E2/M1)|=0.834 $\pm$ 0.210) was determined. Nonzero value of $δ$(E2/M1) raises a question about the existing theoretical interpretation of low-lying levels of $^{227}$Th.
The inelastic scattering cross section for muons, $μ^-$, with energies $E$=9--100~eV from the $^{229}$Th nuclei is calculated in the framework of the second order of the perturbation theory for the quantum electrodynamics. The dominant contribution to the excitation of the low energy isomer $^{229m}$Th$(3/2^+,8.19\pm0.12$~eV) comes from the $E2$ multipole. The excitation cross section reaches the value of $10^{-21}$ cm$^2$ in the range $E\approx$10~eV. This is four to five orders of magnitude larger than the electron excitation cross section and enough for efficient excitation of $^{229m}$Th on the muon beam at the next generation of muon colliders.
Tomas Sikorsky, Jeschua Geist, Daniel Hengstler
et al.
We present a measurement of the low-energy (0--60$\,$keV) $γ$ ray spectrum produced in the $α$-decay of $^{233}$U using a dedicated cryogenic magnetic micro-calorimeter. The energy resolution of $\sim$$10\,$eV, together with exceptional gain linearity, allow us to measure the energy of the low-lying isomeric state in $^{229}$Th using four complementary evaluation schemes. The most accurate scheme determines the $^{229}$Th isomer energy to be $8.10(17)\,$eV, corresponding to 153.1(37)$\,$nm, superseding in precision previous values based on $γ$ spectroscopy, and agreeing with a recent measurement based on internal conversion electrons. We also measure branching ratios of the relevant excited states to be $b_{29}=9.3(6)\%$ and $b_{42}=0.3(3)\%$.
Excitation of the anomalously low lying nuclear isomer $^{229m}$Th$(3/2^+, 8.28 \pm 0.17$ eV) in the process of inelastic electron scattering is studied theoretically in the framework of the perturbation theory for the quantum electrodynamics. The calculated cross sections of $^{229m}$Th by the extremely low energy electrons in the range 9 eV--12 eV for the Th atom and Th$^{1+,4+}$ ions lie in the range $10^{-25}$--$10^{-26}$ cm$^2$. Being so large, the cross section opens up new possibilities for the effective non-resonant excitation of $^{229m}$Th in experiments with an electron beam or electron (electric) current. This can be crucial, since the energy of the isomeric state is currently known with an accuracy insufficient for the resonant excitation by photons. In addition, the cross section of the time reversed process is also large, and as a consequence, the probability of the non-radiative $^{229m}$Th decay via the conduction electrons in metal is $\approx10^{6}$ s$^{-1}$, that is, close to the internal conversion probability in the Th atom.
Benedict Seiferle, Lars von der Wense, Pavlo V. Bilous
et al.
The first nuclear excited state of $^{229}$Th offers the unique opportunity for laser-based optical control of a nucleus. Its exceptional properties allow for the development of a nuclear optical clock which offers a complementary technology and is expected to outperform current electronic-shell based atomic clocks. The development of a nuclear clock was so far impeded by an imprecise knowledge of the energy of the $^{229}$Th nuclear excited state. In this letter we report a direct excitation energy measurement of this elusive state and constrain this to 8.28$\pm$0.17 eV. The energy is determined by spectroscopy of the internal conversion electrons emitted in-flight during the decay of the excited nucleus in neutral $^{229}$Th atoms. The nuclear excitation energy is measured via the valence electronic shell, thereby merging the fields of nuclear- and atomic physics to advance precision metrology. The transition energy between ground and excited state corresponds to a wavelength of 149.7$\pm$3.1 nm. These findings set the starting point for high-resolution nuclear laser spectroscopy and thus the development of a nuclear optical clock of unprecedented accuracy. A nuclear clock is expected to have a large variety of applications, ranging from relativistic geodesy over dark matter research to the observation of potential temporal variation of fundamental constants.
P. V. Borisyuk, E. V. Chubunova, N. N. Kolachevsky
et al.
The results of experimental studies of the low-energy isomeric state in the $^{229}$Th nucleus are presented. The work is consisted of several stages. During the first stage $^{229}$Th nuclei were excited with the inverse internal conversion to the low-lying isomeric level in plasma that was formed by laser pulse at the $^{229}$Th-containing target surface. Then thorium ions having excited nuclei were extracted from the plasma by an external electrical field and implanted into thin SiO$_2$ film grown on a silicon substrate (that is a dielectric material with about 9 eV band-gap). Gamma decay of isomeric nuclei was registered during the second stage by the general methods of the electron spectroscopy after the photon-electron emission from the silicon substrate. Substitution of the photon registration with the electron one allowed us to increase the desired signal by several orders of magnitude and detect the $^{229}$Th nuclei decay. During the third stage the electron spectra from standard Xe VUV source was obtained that allowed determining the energy of photons. In order to prove that the detected signal is caused by isomeric $^{229}$Th nuclei decay a series of experiments was carried. The analysis of electron spectra gives the following results: the energy of the nuclear transition is $E_{\text{is}}=7.1(^{+0.1}_{-0.2})$~eV, the half-life of the isomeric level in bare nucleus in vacuum is $T_{1/2}=1880\pm170$~s, the reduced probability of the isomeric nuclear transition is $B_{\text{W.u.}}(M1;3/2^+\rightarrow 5/2^+)=(3.3\pm0.3)\times10^{-2}$.