# Isotope dating ea equation

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Theoretical nuclear physics Abstract Second-order processes in physics is a research topic focusing attention from several fields worldwide including, for example, non-linear quantum electrodynamics with high-power lasers, neutrinoless double-β decay, and stimulated atomic two-photon transitions.

For the electromagnetic nuclear interaction, the observation of the competitive double-γ decay from mBa has opened up the nuclear structure field for detailed investigation of second-order processes through the manifestation of off-diagonal nuclear polarisability.

Here, we confirm this observation with an 8. Our results, however, contradict the conclusions from the original experiment, where the decay was interpreted to be dominated by a quadrupole-quadrupole component.

Here, we find a substantial enhancement in the energy distribution consistent with a dominating octupole-dipole character and a rather small quadrupole-quadrupole component in the decay, hindered due to an evolution of the internal nuclear structure. The implied strongly hindered double-photon branching in mBa opens up the possibility of the double-photon branching as a feasible tool for nuclear-structure studies on off-diagonal polarisability in nuclei where this hindrance is not present.

Download PDF Introduction Polarisability is a fundamental concept in isotope dating ea equation and chemistry defined from the principles of electromagnetic interaction. It describes how applied electric or magnetic fields induce an electric or magnetic dipole, or higher-order multipole, moment in the matter under investigation 1. In nuclear physics, the simple concept of polarisability influences observables over a broad range of topics.

For example, the static dipole polarisation of the shape of the ground and excited states in atomic nuclei is influenced by the coupling to high-energy collective modes like the giant dipole resonance GDR via virtual excitations.

By expanding the concept of polarisability beyond the scalar case, one can divide the polarisability tensor into separate components. Typically, these are either spatial components like the birefringence properties of crystals or electric and magnetic multipole components.

Within the nuclear structure framework, this type of off-diagonal polarisabilities can appear in very weak second-order processes. In the electromagnetic case, the off-diagonal nuclear polarisability can be defined analogously to Eq. In the definition above, the denominator depends on the interference frequency, ω, of the emitted γ rays and is approximated to be half of the initial state energy.

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- Electromagnetic character of the competitive γ γ / γ -decay from m Ba | Nature Communications
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Here single γ-emission is blocked, and only conversion-electron decay and double-γ decay are allowed. In these experiments, the obtained information consists of correlations between energies and angles of these γ-rays, used to determine the decay probabilities of electric and magnetic dipoles.

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Even though unsuccessful in that respect, these experiments successfully measured an E5 transition with the branching of 1. The decay diagram of this process is shown in Fig. Illustration of the single-γ and the two types of double-γ decay, as fed by the β decay of Cs, including half lives of Cs and mBa. The energy of the Cs ground state Qβ is given relative to the Ba ground state.

Here, M4 corresponds to the single-photon decay. The blue and pink decays show the lowest octupole-dipole and quadrupole—quadrupole components, respectively. Full size image Given the nature of this experiment to observe a longstanding prediction of a fundamental concepts in quantum mechanics and quantum electrodynamics, and the possibility to extract nuclear structure observables from this, it is highly desirable to independently confirm this observation.

Some possibilities that have been under discussion to perform this independent confirmation is to either return to the HPGe approach with complex detector systems and event processing like the Advanced GAmma Tracking Array AGATA set-up 1314 or highly charged radioactive ions The experimental set-up was optimised for obtaining a clean signal over a wide angular range 21 based on the reported intensities of the decay mode.

Here, we can confirm the existence of the isotope dating ea equation double-photon decay process in atomic nuclei with an 8. We, however, find a significant octupole-dipole, E3M1, matrix element product contribution to the double-γ decay mode of Ba, contradicting the conclusions of the original experiment This most extreme case is also the case that best reproduces the αE3M1 polarisability.

In this case, experiments would be feasible also with more exotic sources 22or even in-beam experiments within reasonable beam times, to follow the evolution of the quadrupole—quadrupole strength.

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The detector configuration was a circle with an inner radius to the front-face of the scintillators of 40 cm. This distance was enough to separate true coincidences from multiple Compton scattering of single γ rays using the photon time-of-flight TOFsee Fig. The relative angles between the eleven detectors were This gave five independent γγ-correlation angles centred at: The detectors were separated with a minimum of approximately 15 cm of effective lead shielding between two neighbouring detectors to remove any isotope dating ea equation from single Compton scattering between detector pairs at low angles.

The set-up was characterised both with an in-house toolkit 23 based on the GEANT4 framework 24and a Eu source with an activity of kilo Isotope dating ea equation kBq and a 60Co source with an activity of 60 kBq. For a comprehensive overview, see reference The source had a thin circular active area with a diameter of 3 mm, encapsulated in the centre of a cylindrical polymethylmethacrylate capsule with a diameter of 25 and 3 mm thickness, see Fig.

The location and geometry of the source are also shown, with the active area marked in black.

The time condition for prompt γ-rays is shown as red dashed lines and verified with a Eu source. Corrections for detector efficiencies were done on an event-by-event basis The full data set, as well as the different angular groups, were used to construct the summed energy spectra. The peaks were fitted assuming a quadratic background both with a Gaussian distribution as well as GEANT4 simulated data. Both fitting methods gave consistent results.

Black data points show the summed energy of two coincident photons detected in the CeBr3 detectors for events with a multiplicity of two.

Grey data points show the sum energy spectrum when the multiplicity is larger than two, which mainly correspond to the background induced by cosmic ray showers. We also show the fit to the data of a quadratic background as a dashed red line and the fit of the background plus a Gaussian peak as a solid red line. The error bars represent the one standard-deviation statistical uncertainty. Given an angle, θ1,2, the differential decay is integrated over the frequency of the γ ray, ω.

The frequency is proportional to the energy, and the integration limits are taken as the edges of the energy bin of interest. In the experimental spectrum, a natural low-energy limit comes from the low-energy threshold of the detectors around keV.

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The upper limit was chosen as half of the total energy as we are not able to distinguish any relative ordering of the γ rays. This procedure was performed for all combinations of θ1,2 and δ was evaluated as a function of angle.

The results of this evaluation are shown in Fig. The angular isotope dating ea equation of the two photons emitted in the double-γ decay from this work and reference 11compared with the expected angular distributions of pure M2E2 and E3M1 decay. Full size image This data can be directly fitted to the generalised polarisability functions of Eq.

Other components like αE2M2 or αM3E1 could, in principle, also contribute. However, the general polarisability functions are linearly dependent in the exchange of terms, weighted by the coefficients given by the Wigner 6j symbols, and this experiment is not sensitive to this ordering. These additional components are, furthermore, expected to be small. Thus, we restrict the discussion to the αM2E2 and αE3M1 polarisabilities from here on.

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Energy-sharing distributions The angular distributions themselves are not enough to completely distinguish between the contribution from the different polarisabilities. When calculating the goodness-of-fit χ2two local minima corresponding to either a large αM2E2 component or a large αE3M1 component appear.

Instead, it is necessary to study the energy-sharing distributions between the two individual γ-rays. From Eqs. For this purpose, δ from Eq. Figure 5 a shows the results of these evaluations. When including the data from Walz et al.

The resulting χ2 surface is shown in Fig. As seen here, the χ2 analysis from this data favours a large αE3M1 component, in contradiction with both the experimental interpretation and theoretical conclusions reported in ref.

The data points correspond to the sum of the differential branching ratio defined in Eq. The contours are separated by one standard deviation.

The data from the present work is shown as a black point, data from reference 11 is shown as a red point, and a fit with the two data sets combines is shown as a purple point.

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The error bars represent the one standard-deviation statistical uncertainty, except the error bars in Eγ,low that represent the width of the energy bin. Full size image Discussion To understand these results, we performed theoretical calculations of the polarisation functions isotope dating ea equation Eq. The application of the QPM in the case of odd-mass spherical nuclei is discussed in isotope dating ea equation in reference In particular, the nuclear structure of Ba was studied within the framework of this model in references 2728 and in reference In the work presented here, the calculations were built on the EDF theory coupled with the QPM 29 to obtain magnetic and electric spectral distributions.

To understand the origin of this discrepancy, the properties of the dominant, low-lying, states were investigated from another perspective using the state-of-the-art nuclear MCSM 33 This can explain the observed suppression of αE2M2. Table 2 lists the contributing low-lying matrix elements discussed here.

### Introduction

Isotope dating ea equation 2 Calculated matrix elements. The digitisers were controlled using the Multi Instance Data Acquisition System software and triggered individually. Each event consisted of the energy, the time-stamp, and the the digitised voltage pulse from the detector.

The sub-nanosecond time information was obtained from the value of the time-stamp corrected by a digital interpolation of the sampling points in the recorded pulse, at half of the maximum value of the pulse and interpolated using a quadratic polynomial.

The general polarisability functions in Eq. The quasiparticle-phonon model The QPM Hamiltonian includes mean field, pairing interaction and separable multipole and spin-multipole interactions