In physical chemistrythe Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius inbased on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in that van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions.

This equation has a vast and important application in determining rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula. The Eyring equationdeveloped inalso expresses the relationship between rate and energy. Arrhenius equation gives the dependence of the rate constant of a chemical reaction on the absolute temperaturea pre-exponential factor and other constants of the reaction.

The only difference is the energy units of E a : the former form uses energy per molewhich is common in chemistry, while the latter form uses energy per molecule directly, which is common in physics. The different units are accounted for in using either the gas constantRor the Boltzmann constantk Bas the multiplier of temperature T. The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction.

It can be seen that either increasing the temperature or decreasing the activation energy for example through the use of catalysts will result in an increase in rate of reaction. Given the small temperature range of kinetic studies, it is reasonable to approximate the activation energy as being independent of the temperature. Taking the natural logarithm of Arrhenius equation yields:. This procedure has become so common in experimental chemical kinetics that practitioners have taken to using it to define the activation energy for a reaction.

The modified Arrhenius equation [6] makes explicit the temperature dependence of the pre-exponential factor. The modified equation is usually of the form. Theoretical analyses yield various predictions for n. Another common modification is the stretched exponential form [ citation needed ].

This is typically regarded as a purely empirical correction or fudge factor to make the model fit the data, but can have theoretical meaning, for example showing the presence of a range of activation energies or in special cases like the Mott variable range hopping.

Arrhenius argued that for reactants to transform into products, they must first acquire a minimum amount of energy, called the activation energy E a.

## 0nubetabeta-decay nuclear matrix elements with self-consistent short-range correlations

At an absolute temperature Tthe fraction of molecules that have a kinetic energy greater than E a can be calculated from statistical mechanics. The concept of activation energy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories. One approach is the collision theory of chemical reactions, developed by Max Trautz and William Lewis in the years — In this theory, molecules are supposed to react if they collide with a relative kinetic energy along their line of centers that exceeds E a.

The number of binary collisions between two unlike molecules per second per unit volume is found to be [7]. Here P is an empirical steric factoroften much less than 1, which is interpreted as the fraction of sufficiently energetic collisions in which the two molecules have the correct mutual orientation to react. The Eyring equationanother Arrhenius-like expression, appears in the " transition state theory " of chemical reactions, formulated by WignerEyringPolanyi and Evans in the s.

The Eyring equation can be written:. At first sight this looks like an exponential multiplied by a factor that is linear in temperature.Exponential functions tell the stories of explosive change. The two types of exponential functions are exponential growth and exponential decay. Four variables percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period play roles in exponential functions.

Use an exponential decay function to find the amount at the beginning of the time period. Exponential decay is the change that occurs when an original amount is reduced by a consistent rate over a period of time.

If you are reading this article, then you are probably ambitious. Six years from now, perhaps you want to pursue an undergraduate degree at Dream University.

After sleepless nights, you, Mom, and Dad meet with a financial planner. Study hard. If you prefer to rewrite the equation with the constanton the right of the equation, then do so.

Stick with it! Solve by dividing. Woodforest, Texas, a suburb of Houston, is determined to close the digital divide in its community. A few years ago, community leaders discovered that their citizens were computer illiterate.

They did not have access to the internet and were shut out of the information superhighway. The leaders established the World Wide Web on Wheels, a set of mobile computer stations.

## Arrhenius equation

World Wide Web on Wheels has achieved its goal of only computer illiterate citizens in Woodforest. Community leaders studied the monthly progress of the World Wide Web on Wheels. According to the data, the decline of computer illiterate citizens can be described by the following function:. How many people are computer illiterate 10 months after the inception of the World Wide Web on Wheels?

The variable y represents the number of computer illiterate people at the end of 10 months, so people are still computer illiterate after the World Wide Web on Wheels began to work in the community. How many people were computer illiterate 10 months ago, at the inception of the World Wide Web on Wheels?

If these trends continue, how many people will be computer illiterate 15 months after the inception of the World Wide Web on Wheels?

Share Flipboard Email. Jennifer Ledwith. Math Expert. Jennifer Ledwith is the owner of tutoring and test-preparation company Scholar Ready, LLC and a professional writer, covering math-related topics. Updated October 13, Here's an exponential decay function:.

### Short-range correlations and neutrinoless double beta decay

This function describes the exponential growth of the investment:. Do not solve this exponential equation by dividingby 6. It's a tempting math no-no. Use order of operations to simplify. Freeze: You're not done yet; use order of operations to check your answer. Compare this function to the original exponential growth function:.

Does this function represent exponential decay or exponential growth?Neutrinoless double-beta decay is a predicted beyond Standard Model process that could clarify some of the not yet known neutrino properties, such as the mass scale, the mass hierarchy, and its nature as a Dirac or Majorana fermion. Should this transition be observed, there are still challenges in understanding the underlying contributing mechanisms.

We perform a detailed shell model investigation of several beyond Standard Model mechanisms that consider the existence of right-handed currents. Our analysis presents different venues that can be used to identify the dominant mechanisms for nuclei of experimental interest in the mass region Sn, Te, and Xe.

It requires accurate knowledge of nine nuclear matrix elements that we calculate in addition to the associated energy-dependent phase space factors. Should the neutrinoless double-beta decay be experimentally observed, the lepton number conservation is violated by two units and the back-box theorems [ 1 — 4 ] predict the neutrino to be a Majorana particle.

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In addition to the nature of the neutrino whether a Dirac or a Majorana fermionthere are other unknown properties of the neutrino that could be investigated viasuch as the mass scale, the absolute mass, or the underlying neutrino mass mechanism.

There are several beyond Standard Model mechanisms that could compete and contribute to this process [ 56 ]. Reliable calculations of the nuclear matrix elements NME are necessary to perform an appropriate analysis that could help evaluate the contribution of each mechanism.

The most commonly investigated neutrinoless mechanism is the so-called mass mechanism involving the exchange of light left-handed neutrinos, for which the NME were calculated using many nuclear structure methods. Calculations that consider the contributions of heavy, mostly sterile, right-handed neutrinos have become recently available, while left-handed heavy neutrinos have been shown to have a negligible effect [ 78 ] and their contribution is generally dismissed.

A comparison of the recent mass mechanism results obtained with the most common methods can be seen in Figure 6 of [ 9 ], where one can notice the differences that still exist among these nuclear structure methods.

Figure 7 of [ 9 ] shows the heavy neutrino results for several nuclear structure methods, and the differences are even larger than those in the light neutrino case because of the uncertainties related to the short-range correlation SRC effects. There are efforts to reduce these uncertainties by the development of an effective transition operator that treats the SRC consistently [ 10 ].

Because shell model calculations were successful in predicting two-neutrino double-beta decay half-lives [ 11 ] before experimental measurements and as shell model calculations of different groups largely agree with each other without the need to adjust model parameters, we calculate our nuclear matrix elements using shell model techniques and Hamiltonians that reasonably describe the experimental spectroscopic observables. Experiments such as SuperNEMO [ 1213 ] could track the outgoing electrons and help distinguish between the mass mechanism and so-called and mechanisms [ 1415 ].

This would also provide complementary data at low energies for testing the existence of right-handed contributions predicted by left-right symmetric models [ 15 — 19 ], currently investigated at high energies in colliders and accelerators such as LHC [ 20 ]. To distinguish the possible contribution of the heavy right-handed neutrino using shell model nuclear matrix elements, measurements of lifetimes for at least two different isotopes are necessary, ideally that of an isotope and another lifetime of an isotope, as discussed in Section V of [ 21 ].

It is expected that if the neutrinoless double-beta decay is confirmed in any of the experiments, more resources and upgrades could be dedicated to boost the statistics and to reveal more information on the neutrino properties. Following our recent study for 82 Se in [ 21 ], which is the baseline isotope of SuperNEMO, we extend our analysis of and mechanisms to other nuclei of immediate experimental interest: Sn, Te, and Xe.

These isotopes are under investigation by the TIN. For the mass regionwe perform calculations in model space consisting of and valence orbitals using the SVD shell model Hamiltonian [ 29 ] that was fine-tuned with experimental data from Sn isotopes. Calculations of NME in larger model spaces e. In this work, assuming the detection of several tens of decay events, we present a possibility to identify right-handed contributions from and mechanisms by analyzing the two-electron angular and energy distributions that could be measured.

We organize this paper as follows: Section 2 shows a brief description of the neutrinoless double-beta decay formalism considering a low-energy Hamiltonian that takes into account contributions from right-handed currents. Section 3 presents an analysis of the half-lives and of the two-electron angular and energy distributions results for Sn, Te, and Xe.

Finally, we dedicate Section 4 to conclusions. One model that considers the right-handed currents contributions and includes heavy particles that are not part of the Standard Model is the left-right symmetric model [ 1718 ]. Within the framework of the left-right symmetric model, the neutrinoless double-beta decay half-life expression is where, and are neutrino physics parameters defined in [ 15 ] see also Appendix A of [ 21 ]and are the light and heavy neutrino-exchange nuclear matrix elements [ 56 ], and and are combinations of NME and phase space factors, which are calculated in this paper.

We consider a seesaw type I dominance [ 43 ] and we will neglect this contribution here. For an easier read, we perform the following change of notation:, and. In this paper, we provide an analysis of the two-electron relative energy and angular distributions for Sn, Te, and Xe using shell model NME that we calculate. The purpose of this analysis is to identify the relative contributions of and terms in 1.

As in [ 21 ], we start from the classic paper of Doi et al. By simplifying some notations and ignoring the contribution from term, which has the same energy and angular distribution as term, the half-life expression [ 14 ] is written as where and are the relative CP-violating phases Eq.

These are defined as where are combinations of nuclear matrix elements and phase space factors PSF. Their expressions can be found in Appendix B, Eqs. C2 and C3 of [ 34 ], respectively. We can express the half-life as follows: with the normalized kinetic energy defined as where is the -value of the decay.Box 35, Jyvaskyla, Finland.

A comprehensive analysis of the structure of the nuclear matrix elements NMEs of neutrinoless double beta-minus decays to the ground and first excited states is performed in terms of the contributing multipole states in the intermediate nuclei of transitions. We concentrate on the transitions mediated by the light l-NMEs Majorana neutrinos. As nuclear model we use the proton-neutron quasiparticle random-phase approximation pnQRPA with a realistic two-nucleon interaction based on the Bonn one-boson-exchange matrix.

In the computations we include the appropriate short-range correlations, nucleon form factors, and higher-order nucleonic weak currents and restore the isospin symmetry by the isoscalar-isovector decomposition of the particle-particle proton-neutron interaction parameter. Thanks to neutrino-oscillation experiments much is known about the basic properties of the neutrino concerning its mixing and squared mass differences. What is not known is the absolute mass scale, the related mass hierarchy, and the fundamental nature Dirac or Majorana of the neutrino.

This can be studied by analyzing the neutrinoless double beta decays of atomic nuclei [ 1 — 4 ] through analyses of the participating nuclear matrix elements NMEs.

The decays proceed by virtual transitions through states of all multipoles in the intermediate nucleus, being the total angular momentum and being the parity of the intermediate state.

Most of the present interest is concentrated on the double beta-minus variant decay of the decays due to their relatively large decay energies values and natural abundancies. In this work we concentrate on analyses of the intermediate contributions to the decays for the ground-state-to-ground-state and ground-state-to-excited-state transitions in nuclear systems of experimental interest.

We focus on the light Majorana neutrino mediated transitions by taking into account the appropriate short-range nucleon-nucleon correlations [ 5 ] and contributions arising from the induced currents and the finite nucleon size [ 6 ].

There are several nuclear models that have recently been used to compute the decay NMEs see, e. However, the only model that avoids the closure approximation and retains the contributions from individual intermediate states is the proton-neutron quasiparticle random-phase approximation pnQRPA [ 712 — 14 ].

Some analyses of the intermediate-state contributions within the pnQRPA approach have been performed in [ 12131516 ] and recently quite extensively in [ 17 ]. In [ 17 ] an intermediate multipole decomposition was done for decays of 76 Ge, 82 Se, 96 Zr, Mo, Pd, Cd, Sn,Te, and Xe to the ground state of the respective daughter nuclei. In this paper we extend the analysis of [ 17 ] to a more detailed scrutiny of the intermediate contributions to the decay NMEs of the above-mentioned nuclei.

We also extend the scope of [ 17 ] by considering transitions to the first excited states in addition to the ground-state-to-ground-state transitions. In this section a very brief introduction to the computational framework of the present calculations is given. The present analyses on ground-state-to-ground-state decays are based on the calculations done in [ 17 ]. Details considering the excited-state decays are given in a future publication.

We assume here that the decay proceeds via the light Majorana neutrino so that the inverse half-life can be written as where is a phase-space factor for the final-state leptons defined here without the axial vector coupling constant. The quantity denotes the neutrino effective mass and describes the physics beyond the standard model [ 17 ]. The quantity is the light neutrino nuclear matrix element l-NME. The nuclear matrix element can be decomposed into Gamow-Teller GTFermi Fand tensor T contributions as where is the vector coupling constant.

The final state,can be either the ground state or an excited state of the daughter nucleus, and the overlap factor between the two one-body transition densities helps connect the corresponding intermediate states emerging from the pnQRPA calculations in the mother and daughter nuclei. As mentioned before, our calculations contain the appropriate short-range correlators, nucleon form factors, and higher-order nucleonic weak currents.

In addition, we decompose the particle-particle proton-neutron interaction strength parameter of the pnQRPA into its isoscalar and isovector components and adjust these components independently as described in [ 17 ]: the isovector component is fixed such that the NME of the two-neutrino double beta-decay vanishes and the isospin symmetry is thus restored for both the and decays.

The isoscalar component, in turn, is fixed such that the measured half-life of the decay is reproduced. The resulting values of both components of are shown in Table I of [ 17 ].Busque entre los mas de recursos disponibles en el repositorio.

The nuclear-structure components of the calculation, that is the participant nuclear wave functions, have been calculated in the shell-model scheme for 48Ca and in the proton-neutron quasiparticle random-phase approximation pnQRPA scheme for 76Ge. We compare the traditional approach of using the Jastrow correlation function with the more complete scheme of the unitary correlation operator method UCOM. JavaScript is disabled for your browser. Some features of this site may not work without it.

Buscar material Busque entre los mas de recursos disponibles en el repositorio. Short-range correlations and neutrinoless double beta decay Autores: Kortelainen, M. Civitarese, Enrique Osvaldo Suhonen, J. Toivanen, J. Tipo de documento: Articulo. Revista: Physics Letters B; vol.

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Otros Identificadores: eids2. ISSN: Palabras claves: Neutrinoless double beta decay ; Nuclear matrix elements ; Short-range correlations ; Unitary correlation operator method. Descargar archivos. Documento completo Descargar archivo Creado el: 10 de octubre de Bibliography tool, search by references, institution filtering.

Learn more. Lukas Graf University Coll. London and Yale U. Frank F. Deppisch University Coll. Francesco Iachello Yale U. Jenni Kotila Jyvaskyla U. Published in: Phys. D 98 9 DOI: Citations per year 5 15 4. Abstract: APS. Note: 42 pages, 9 figures, added discussion of QCD running and decay distributions. Beyond the standard model double-beta decay: 0neutrino lepton number: violation neutrino: Majorana neutrino: mass effective Lagrangian interaction: short-range operator: higher-dimensional angular correlation phase space.

References Figures 9. Georges Aad Freiburg U. B Serguei Chatrchyan Yerevan Phys. Review of Particle Physics K. Olive Minnesota U. Agashe Maryland U. Amsler U. Bern, AEC. Antonelli Frascati. Arguin Montreal U. C 38 Planck results. Cosmological parameters Planck Collaboration.

Ade Cardiff U. Peter Minkowski Bern U. B 67 Mohapatra City Coll. Goran Senjanovic Maryland U.Reliability of HFB wave functions generated with four different parametrizations of the pairing plus multipolar type of effective two-body interaction has been ascertained by comparing a number of nuclear observables with the available experimental data. Effects due to different parametrizations of effective two-body interactions, form factors and short-range correlations have been studied.

Uncertainties in NTMEs calculated with wave functions generated with four different parametrizations of the pairing plus multipolar type of effective two-body interaction, dipole form factor and three different parametrizations of Jastrow type of short-range correlations within mechanisms involving light Majorana neutrinos, heavy Majorana neutrinos, sterile neutrinos and Majorons have been statistically estimated.

The story of neutrinos as well as weak interaction is rather checkered and quite exciting due to their enigmatic nature. In addition, information on the lepton number violation, possible hierarchies in the neutrino mass spectrum, the origin of neutrino mass and CP violation in the leptonic sector can be inferred [ 34 ].

The main objective of the nuclear structure calculations is to provide reliable nuclear transition matrix elements NTMEs for extracting gauge theoretical parameters with minimum uncertainty. In the conventional nuclear models, there are three basic ingredients, namely the model space, the single particle energies SPEs and the effective two body interaction.

Usually, they are chosen on the basis of practical considerations albeit a consistent procedure is available for their choice. In spite of the impressive success of shell-model approach [ 13 — 26 ], presently available numerical facilities highly limit its application to the description of medium and heavy mass nuclei. In the shell model [ 16 — 18 ], the effect of deformation on the NTMEs has been investigated and it has been shown that the NTMEs are the largest for equal deformation of parent and daughter nuclei.

However, different predictions are obtained for other observables. With the consideration of two models, namely QRPA and RQRPA, three sets of basis states and three realistic two-body effective interactions based on the charge dependent Bonn, Argonne, and Nijmen potentials, theoretical uncertainties have been estimated by Rodin et al.

Further, the approach of Rodin et al. Recently, it has been suggested by Engel [ 59 ] that the systematic errors can be reduced by including all physics known to be important in the error analysis following Reference [ 60 ]. In Rath et al. With the assumption that the nucleus consists of non-relativistic point nucleons and neglecting many-nucleon forces, the conventional nuclear many-body Hamiltonian H in an appropriate model space M is given by.

An effective operator O eff is also to be defined such that the same observable are obtained in a finite dimensional model space M as reproduced by a bare operator O acting in an infinite dimensional Hilbert space. The problems associated with the numerical implementation of such effective operators in perturbative approach have been discussed by Haxton and Lau [ 81 ]. In the HFB theory [ 82 ], the HF field and the pairing interaction are treated simultaneously and on equal footing.

Further, the particle coordinates are transformed to quasiparticle coordinates through general Bogoliubov transformation. Essentially, the Hamiltonian H is expressed as. In the HFB theory, the interaction between the quasiparticles is usually neglected and the Hamiltonian H is approximated by independent quasiparticle Hamiltonian.

Specifically, the wave functions are obtained by minimizing the expectation value of the effective Hamiltonian consisting of single particle Hamiltonian H spand pairing V P plus multipolar quadrupole-quadrupole V QQ and hexadecapole-hexadecapole V HH parts type of effective two-body interaction [ 75 ].

Table 1. Notably, this expression is same as reported by Hirsch et al. Presently, each proton-neutron excitation is treated according to its spin-flip or non—spin-flip nature and the spin-orbit splitting is explicitly included in the energy denominator. Hence, the use of summation method goes beyond the closure approximation and previously employed summation method in the pseudo SU 3 model [ 9092 ].

In Chandra et al. Table 2.

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