Molecular final state energies and transition probabilities have been computed for beta-decay of the tritium molecule. The results are of sufficient accuracy to make a determination of the electron neutrino rest mass with an error not exceeding a few tenths of an electron volt. Effects of approximate models of tritium beta-decay on the neutrino mass determination are discussed. 14 refs., 3 figs., 1 tab

Fackler, O.

Jeziorski, B.

Kolos, W.

Monkhorst, H.J.

Mugge, M.

Szalewicz, K.

UNT Digital Library

UCRL—94429 DE86 009408 Molecular Effects in the Neutrino Mais Determination Froa Beta-Decay of The Tritium Molecule O.Fackler"' ,B.Jeziorskic,W.Kolosc, H. J.Monkhorst, M.Mugge* and K.Szalewiczc a Lawrence Llvermore National Laboratory, University of California, Liveraore, California 94550, USA b Department of Physics, The Rockefeller University, New York, New York 10021, USA c Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, 02-093 Warsaw, Poland d Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611, USA Abstract Molecular final state energies and transition probabilities have been computed for beta-decay of the tritiun molecule. The results are of sufficient accuracy to make a determination of the electron neutrino rest mass with an error not exceeding a few tenths of an electron volt. Effects of approximate models of tritium beta-decay on the neutrino mass determination are discussed AMSIB? DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED^ I. Introduction Experiments are being prepared or performed in several laboratories aiming at electron neutrino mass determination from measurements of the tritium /9-decay spectrum near the endpoint. These experiments can be separated into (1) those dealing with atomic, molecular or solid state tritium, and (2) those in which the decaying tritons have a more complex molecular or solid state surrounding. OnJy in the first case can one perform reliable quantum mechanical computations for the final states of the daughter system, and therefore only in these experiments will the i accuracy of the neutrino mass determination depend solely on the experimental resolution. In the second group of experiments the inherently less accurate theoretical results for the final states will always introduce an additional uncertainty into the determined value of the neutrino mass. In the present paper we give a summary of our accurate theoretical treatment of molecular effects accompanying the /9-decay of T 2. Ue consider transitions to the electronic ground state of HeT , to excited bound states of this ion, »=. well as to the resonance and scattering states. In the case of transitions to lowest electronic states we explicity consider excitations to all possible rovibrational (rotational and vibrational) states. These results are used to construct the Kurie plots and to perform numerical experiments that show the magnitude of error in the neutrino mass determination arising if the accurate treatment of the final states is replaced by a more approximate one. He also discuss the solid state effects in frozen T 2 and show that they are either small or computable. The results presented below will be used to interpret the data from the frozen tritium experiment now under way at LLNL (see the accompanying contribution in these proceedings). II. Kurie plots and their accuracy It is well known that to extract the neutrino rest mass from the ft-« decay spectrum one must compute the Kurie plots defined as vE ; w - (\+ E - v^itv+ E • v 2 • " 2 ] 1 / 4 ( 2> where 2 denotes the 0-electron energy relative to the endpolnt energy, m the neutrino rest aass, P the transition probability to the n-th final state of thi' daughter ion, and E the energy of this final state measured relative to the lowest state energy. The process under consideration is T 2 -» He + + e" + v 1 + e l Since the ground state of T 3 is a S state, n in Eq.(l) labels all Z rovibronic (rotational, vibrational and electronic) states of the He . The values of E_ and P for all these states are needed to calculate the n n Kurie plots (1). The following basic methods and techniques have been applied in our work to provide the necessary input for the Kurie plots: 1. First-order tine-dependent perturbation theory, 2. Neglect the 0-electron interaction with the molecular electrons, 3. Adiabatic approximation, 4. Expansion of the electronic wavefunction in elliptic coordinates, 5. Numerical solution of the vibrational equation. Since we do not use any new approaches we will not present the details of the abovs steps 1-5, and only their accuracy and reliability will be discussed: 1. The applicability of first-order perturbation theory follows from the snallness of the weak interaction operators. 2. In this Born approximation one essentially neglects the following effects: (a) screening of the active nucleus and the final state dependence of this screening, (b) exchange of the ^-electron with the molecular ones, (c) correlation of the ^-electron in its continuum orbital with the bound state electrons. Since in the case of tritium decay the momentum of the 0-electron is relatively large, all three effects should be small, with 7 effect (a) being probably the largest one. It has been shown that for the two lowest states of the T atom the effects (a) and (c) change the transition probability by only 0.17%. There is no reason to suppose that the molecular binding in T 3 may change this value significantly. 3. The adiabatic approximation consists of factoring the molecular wavefunction into electronic and rovibrational wavefunctions. For the ' + 8 ground states of T 2 and HeT one may expect the relative value of the .« nonadiabatic effects in the energy to be less than 10 They may be larger for some higher excited states but since the transition -2-probabilities to these states are considerably saaller than to the ground state we conclude that the adlabatic approximation Is very reliable. 4. The expansion of the electronic *avefunction In terms of the basis set expressed In elliptic coordinates and depending explicitly on the interelectronic distance <r",, with n < 4) is known to b* very rapidly converging (see Kef.l and references therein). Host of our computations have been carried out with 200-tarm expansions and the results indicate that for a few lowest bound states the error in the probabilities is approximately 0.00001. The error may be larger for high lying Rydberg states and also for tha resonance states. .We have shown, however, by performing numerical experiments, that this affect has no effect on our final results relevant to tha neutrino mass determination. 5. The vibrational equation has been solved numerically for all bound 2 rovibrational and continuum states resulting from the nuclear motion in the electronic states 1-4 and 6. Using sum rules we have shown that for > + each considered electronic state of HeT , and a given rotational quantum number j, the error in the transition probabilities summed over the whole spectrum amounts to less than 0.0001. Concluding this section we can say that in our computation of the Kurie plots for the T 2 aolecule the largest error arises from approximations 1 and 2, but not from molecular computations. In our opinion, the accuracy of the Kurie plots for molecular T 2 computed in the present work is equally as accurate as that which can be calculated for atomic T. III. Computation of wavefunctions and of transition probabilities For the electronic ground state of T, we have used the previously • 10 determined wavefunctions * different for different regions of the internuclear distance. For HeT six lowest electronic states have been considered explicitly. For each of these states a selection of terms in the expansion of the electronic wavefunction has been made and the nonlinear parameters have been optimized. Transition probability to the electronic ground state of HeT , computed for the equilibrium internuclear distance in T 3, R - 1.4 bohr, amounts to 58.2%, and to the five lowest excited states 26.4%. By diagonalizing the Hamiltonian matrix we get, however, more eigenvalues below the ionization continuum. Uith our best 249-term wavefunction we obtained 16 states in this region. Transition probabilities to the 10 highest of these states, for which the nonlinear parameters in the wavefunctions have not been individually optimized, sums -3-to 1.1%. Thus the total transition probability to all bound excited states Is 27.5%. For resonance states the energies have been computed using the stabilization aethod. The reliability of this method has been demonstrated by performing (In elliptic coordinates) confutations for the united atom as well as for the separated atoms, and reproducing the 4 s corresponding atomic resonance energies. ' The transition probability to all resonance and scattering states In the continuum amounts to It.3%. In our' most accurate computations we have used 249 terms in the expansion of the H matrix, and the sum of transition probabilities to all these 249 states amounts to 1.00000. Some of the computed potential energy curves are shown in Fig.l. The equilibrium lnternuclear distance for T 2 is 1.4 bohr, and due to the slow motion of , the nuclei the transitions to electronic states of HeT occur without any change of R (Franck-Condon-type transitions). It is seen in Fig.l that only transitions to the ground state can end up in a stable rovibrational state of the daughter molecular ion. For all other states the 2 4 6 _ ,. . . internuclear distance R -R (bohr) 1.4 bohr corresponds to a Fig. 1. Potential energy curves for HeT the various -4-strongly repulsive part of the potential energy curve and no stable vibrational states are possible. To compute the transition probabilities to the rovibrational states in the electronic states of HeT + we had to solve the rovibrational Schrodinger equation d 2f 55- + 2/4E - U n j(R)lf(R) - 0 (3) where U ,(R) denotes the potential governing the nuclear notion and n is the reduced aass of the nuclei. Equation (3) is obtained if the adiabatic, i.e. factorized forn of the molecular wavefunction is assumed * - * n ( ? > ' ? » ; R ) f n v j ( R ) R * l Y j » < , , * ) W In the above wavefunction * (r,,r,;R) denotes the electronic wavefun on computed for the fixed internuclear distance R, Y._(»,(») is the rotational wavefunction and f , (R) describes the vibrational notion. The dependence of th« vibrational wavefunction on the rotational quantum number j arises from the centrifugal term in the nuclear notion potential U . appearing in Eq.(3). This potential has the forn "njW-S-*- + C<R> + U n <R> + U n < R> < 5> where Ir (R) denotes the Born-Oppenheimer (or clamped nuclei) potential for the n-th electronic state, obtained by solving the electronic Schrodinger .ad » equation, UT (R) is the diagonal correction for nuclear motion, and rel » U (R) is the relativistic correction to the potential which is small and has been included only for T 2. If the direction of the emitted ^-electron does not coincide with the direction of the nolecular axis, rotation of the molecule is excited. This rotation gives rise to the centrifugal potential barrier in the potential, leading to rotational dissociation from the quasibound vibrational and scattering states whose energies are above the dissociation limit. In the s + electronic ground state of HeT there are 300 stable rovibrational states. Transition probabilities to all rovibrational states and to the region of 2 the continuum spectrum have been computed and the resulting probability histogram is shown if Fig.2. The ground state dissociation limit is at E -0. Thus transitions to the E > 0 region include transitions to the auasibound and continuum states. It follows from the histogram that they -5-occur with a probability of 16.57%, which Is only smallar by a factor of 2.5 than tha transition probability to tha bound rovibrational statas, tha lattar amounting to 40.84%. 0.06 0.04 -2 0.02 --2.0 EfeV) Fig.2. Probability histogram for transitions to the ground 3 + electronic state of HeT . -6-IV. Raaults and discussion Tha bast probability distribution, computed as described in th* pravious saction, has baan usad to calculata tha Kurla plots . In Fig.3 wo show sons Kurla plots obtainad for a - 30 aV using various modal* of tha dacaying system. Tha curvas labalad "nuclear" and "atonic" show tha rasults obtainad for bara trlton and for tha T aton, raapactivaly. Tha ranainlng resulr.s ara for oolacular T 2. Tha T a(R - l.A) curve shows rasults conputsd for tha aquilibriun intarnuclaar dlstanca R - 1.4 bohr. The "R-avaragad" result* which on tha graph ara indistinguishable fron tha previous one, have bean obtained by averaging the electronic transitions over the zero-point vibrations in the initial state of T2'. The "T2(final)" curve has been computed taking explicitly into account all transitions to €[.V] Fig. 3. Comparison of pure nuclear, atomic, and nolecular T 2 Kurie plots for tritium 0 decay and for m - 30 eV. -7-rovlbratlonal and continuum states In ttu lowest electronic states of H«T +. These transitions considerably change the Kurle plot near the and point, and may have a large effect on the determined value of the neutrino mass If the latter Is small, l.a. of the order of 1 eV or less. Ve have also studied the affect of various models of tritium £-decay and of the experimental resolution on tha determined value of the neutrino mass. The following procedure was employed: We have used our best probability function, assumed m - 30 aV or m^ - 1 eV, and the spectrum thus obtained was convoluted with two different resolution functions. Next we generated the "experimental" spectrum by the Monte Carlo method, fitted the spectrum with various models, and for aach model we determined m^. The results are shown in Table 1. It Is seen that 'they are sensitive to the resolution function, and if m - 30 aV one can expect that any reasonable molecular model would give fairly reliable results. The entry GTO CI in Table 1 corresponds to a configuration interaction type 12 vavefunction in which Gaussian type orbitala have been used This wavefunction gives considerably worse energies for HeT as compared with our values but, in spite of that, it would lead to a fairly accurate value of a.,. This situation changes considerably for n - 1 aV. In this case very accurate results are needed to derive an m value with, say, 50% accuracy. One may expect that if m equals a few eV the situation will be less optimistic than for m - 30 eV, and fairly accurate Kurie plots will be needed for a reliable determination of m . The results presented above refer to molecular T 2. The question arises, how big are the solid state effects that will have to be taken into account to extract the m value from the LLNL frozen tritium experiment. Five such effects can be considered: 1. The fractional change of the T 2 wavefunction, A*/*, caused by the binding in the crystal lattice. This is a small effect and amounts to is < 0.1* in the wavefunction. 2. Perturbation due to the interaction of the HeT ion (treated as a point charge) with the induced dipoles of the surrounding T 2 molecules results in less than an 0.8 eV shift of all excitation energies . In the lowest approximation this energy is independent of the HeT state, and in such a case it affects only the end-point energy but not the value of m . Since in a higher- approximation the interaction depends on the state of HeT , it has some effect on m but being of a higher order this effect can be expected to be small. -8-Table 1. Iffact of various model spectra or neutrino mass Resolution Function ITEP 1984 Gaussian Forward half FWHM - 4.7 aV width 22 «V • An B An v v v v Reference m - 30 aV Bare nucleus 6.4 24 Two-level atomic 25.5 5 Atomic 25.2 5 GTO CI, R - 1.4 bohr 29.2 <1 This work r - 1.4 bohr 30.0 <1 " " R-averaged 30.2 <1 final 29.9 <1 Reference m - 1 eV Bare nucleus Atomic GTO CI, R - 1.4 bohr This voifc R - 1.4 bohr " " R-averaged final 3. Recoilless 0-decay ("Hossbauer effect") has been found to be completely negligible . 4. The sudden charge change of the decaying nucleus may induce excitations not only of the daughter ion but also of the surrounding T 2 molecules. This effect is expected to be small and will be computed in the near future. 5. Multiple scattering of the /^-electron by surrounding T 2 molecules. This effect will be computed in the near future using the methods developed 14 earlier It may be added that the multiple scattering effect is source thickness dependent and its magnitude can be studied experimentally by _varying the thickness of the T 2 layer. 12.3 18 25.4 5 25.3 5 29.3 <1 30.0 <1 30.3 <1 30.2 <1 0.00 1 0.00 1 0.00 1 0.42 Si 1.92 <x 1.00 si -9-V. Conclusions 1. If a is about 30 eV or larger a considerably less sophisticated approach than th*t presented above is sufficient to extract a reliable m value from the experimental data. 2. If m is less than about 1 ev, and the experimental resolution reaches this high degree of accuracy, both the theory and its numerical implementation must be improved. 3. In any case, we can say that results have been obtained needed for a neutrino mass determination with an error not exceeding a few tenths of an aV. This work was supported under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No.W-7405-Eng-48.lt was also supported in part by National Science Foundation Grant No. CHE-82007220, Division of Sponsored Research of the University of Florida, and Polish Academy of Sciences Grant MR.I.9. References 1. W.Kolos, B.Jeziorski, K.Szalewicz and H.J.Monkhorst,Phys.Rev. A 21, 551 (1985). 2. B.Jeziorski, W.Kolos, K.Szalewicz, O.Fackler and H.J.Monkhorst, Phys.Rev. A 12, 2573 (1985). 3. O.Fackler, B.Jeziorski, W.Kolos, H.J.Monkhorst and K.Szalewicz, Phys.Rev.Lett. i5_, 1388 (1985). 4. W.Kolos, B.Jeziorski, H.J.Monkhorst and K.Szalewicz, Intern.J.Quantum Chem. S 12, 000 (1985). 5. O.Fackler, B.Jeziorski, W.Kolos, H.J.Monkhorst and K.Szalewicz, to be published. 6. F.N.D.Kurie, J.R.Richardson and H.C.Paxton, Phys.Rev. 4j>, 368 (1936). 7. R.D.Williams and S.E.Koonin, Phys.Rev. C 21, 1815 (1983). 8. L.Wolniewicz, J.Chem.Phys. JjS, 6173 (1983). 9. W.Kolos end L.Wolniewicz, J.Chem.Phys. 41, 3663 (1964). 10. W.Kolos and L.Wolniewicz, J.Mol.Spectrosc. 54, 303 (1975). 11. H.S.Taylor, G.V.Nazaroff and A.Golebiewski, J.Chem.Phys. 45., 2872 (1966); H.S.Taylor, Adv.Chem.Phys. lfi, 91 (1970). 12. O.Fackler and N.Winter, Unpublished; O.Fackler, M.Mugge, H.Sticker, N.Winter and R. Woerner, Proceedings of the Fourth Horiond Workshop on Massive Neutrinos in Astrophysics and Particle Physics, La Plague, Savoie, France 1984, edited by J.Tran Thanh Van (Edition Frontiers, Paris), p.273. 13. L.Z.Stolarczk and H.J.Monkhorst, unpublished. 14. W.Kolos, H.J.Monkhorst and K.Szalewicz, J.Chem.Phys. 22, 1323, 1335 (1982); K.Szalewicz, W. Kolos, H.J.Monkhorst and C.Jackson, J.Chem.Phys. 80, 1435 (1984). -10-DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United Slates Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use­fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe­cific commercial product, process, or service by trade name, trademark, manufac­turer, or otherwise does not necessarily constitute or imply its endorsement, recom­mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily slate or reflect those of the United States Government or any agency thereof. 

Molecular effects in the neutrino mass determination from beta-decay of the tritium molecule

https://digital.library.unt.edu/ark:/67531/metadc1100302/m2/1/high_res_d/5960888.pdf

Molecular effects in the neutrino mass determination from beta-decay of the tritium molecule

Abstract

Similar works

Full text

Available Versions

UNT Digital Library