Sep 15, 1988 - Continuous electronic spectrum accompanying the P decay of molecular tritium. 0. Hans Agren* and Vincenzo Carravetta. Department of ...
PHYSICAL REVIEW A
Continuous
VOLUME 38, NUMBER 6
electronic spectrum accompanying
the
SEPTEMBER 15, 1988
P decay of molecular tritium
0
Hans Agren* and Vincenzo Carravetta Department
of Chemistry and Minnesota Supercomputer Institute,
University
of Minnesota,
Minnesota 55455 (Received 11 January 1988)
Minneapolis,
The electronic spectrum of the T2~'HeT+ + v+e method using Slater and trigonometric basis functions. at 68 eV the continuum lacks structure.
p decay is obtained by a Stieltjes-imaging Except for one strong resonance structure
METHOD
INTRODUCTION
There has recently been a resurgence of ing P-decay spectra to explore the possibility of a nonzero rest mass for the neutrino. The T2 molecule, either in free or crystalline phase, has provided a prime test unit, experimentally as well as theoretically. In a series of pahave addressed various pers Kolos and co-workers' theoretical and numerical aspects for the use of molecular quantum methods on the P decay of T2 and their implications for a determination of the rest mass of the neutrino. interest in us-
Rotational-vibrational excitations, predissociation, crystalline excitations, and interactions with the P electron, are examples of physical effects with influence on the probability distribution of the daughter ion HeT+ that have been investigated. In the Kurie plots, which form the basic units connecting experimental and theoretical data, the specific molecular effects will have most significance near the end-point energy corresponding to the first few discrete states of the daughter ion. However, the best experimental resolution and intensity seems to be achieved for lower energies which also involve portions of the electronic continuum. An accurate determination of the electronic continuum resonances and cross sections may therefore have a bearing on the neutrino-mass problem. Two theoretical articles specializing on the probability distribution of the electronic continuum following the T2 P decay have also recently appeared. ' The description of continuous electronic spectra pertinent to nonlocal potentials has posed considerable formal and numerical challenges. In recent papers we have demonstrated how Stieltjes-imaging moment theory with square-integrable (L ) basis sets can be implemented for studies of non radiative and photoionization decay shake-oF' processes in molecules. It was shown that L basis sets including one-center expanded Slater and trigonometric functions with high angular momenta describe the electronic continuum excellently over a wide range of energies due to the oscillating nature of such basis sets and due to the amount of control of the redundancy problem associated with their use. Within the framework of the sudden theory, ionization shake-off and P decay constitute equivalent processes, and the previously devised methods can favorably be applied also for the calculation of probability distributions following P decay in light molecules. In the following sections we describe the results of such calculations for the P decay of T2. 38
The comparison of experimental and theoretical Kurie plots provides the key for a determination of a neutrino rest mass from P decay. A theoretical Kurie plot neglecting the Fermi contact interaction displays the function K(e) proportional to the square root of the number of electrons emitted per unit time with energies between c. and a+dc,
K(e;m„)= $, „P„K (e;E„,m, where — c is the energy measured relative to the end point of the spectrum and m, is the energy correspondis the set of ing to the rest mass of the neutrino. discrete and continuous excitation energies of the to populate the daughter ion, and I I the probabilities corresponding levels during the P decay. Each branch of the plot, referring to a specific excitation energy is defined as
[E„)
P„
E„,
E„)'— —m', ]" y [(m, +e —E„)'
K(E;E„,m„)=(m, +E
e(E„.
E)E„,
for and K(e;E„, Thus the m, )=0 for basic effort to produce theoretical Kurie plots lies in the computation of a set of excitation energies and probabilities Within the framework of the Stieltjesimaging moment theory this amounts to first calculating a set of primitive eigenvalues and "oscillator strengths" in a square-integrable basis set. For the discrete part of the spectrum these values provide the true excitation energies, and probabilities, while respectively, Stieltjes imaging is applied to obtain the correct energy normalization in the continuum. This is accomplished by calculating the Stieltjes derivatives
IE„,P„I.
[E„,P„]
n) +P( n) P(i+1 i
d J(n)(E)
dE
2
E.'"', E
(3)
at
of'a cumulative strength function
J(n)(e)
2707
P(n) y E, (E
1988
The American Physical Society
2708
HANS AGREN AND VINCENZO CARRAVETTA
and "oscillator" strengths. Within the sudden approximation for P decay (cf. photoionization shake-oII) we consider a static separation (nonpolarizing and strong orthogonality condition) between primary P (photo-) electron and the residual daughter ion
The discretized spectrum forms generalized quadrature points and weights for the first 2n spectral moments
S( —k)= g P "'(E "') ",
k
=0, 1, 2, . . . , Zn —1
. (6)
S( —k)
are thus obtained directly The various moments from the primitive set of calculated transition energies
HeT+) — Eo( HeT+), (4„(HeT+) P„]=[E„( [E„,
~
'e
)&cos(kr}Y, (e, g)
4( HeT)=%( HeT+)~P(e This assumption
)
.
leads to the primitive eigenset
Vo(Tz))~] .
In contrast to photoionization shake-off in many-electron systems where we obtained the "daughter ion" wave function by means of the static exchange approximation and one-channel approximation for the shake-up/shakeoff functions of the form of Eq. (7), we obtain the wave functions of HeT+ and the ground state of Tz in Eq. (9) of the full Hamiltonian by means of diagonalization within the basis set, i.e., with full configuration interaction (CI). Neglecting all interaction between the P electron and the residual ion, cf. Eq. (I), implies also that we do not consider any contribution to the Kurie plot from the Fermi contact interaction between the nucleus and the P electron. The basis set employed in this work is of the form X„& (g, k, r, 8, $)= A„(g,k)r"
3S
(9)
containing both Slater and trigonometric type functions with high angular momenta. Such basis functions have already been proven to yield excellent continuum orbitals for molecular photoionization processes. The parameter k provides the flexibility for dense representations in the regions with resonances without redundancy problems. Several basis sets have been tested in this work in order to assure stable results and to check the resolution of the Stieltjes imaging; we present here spectra from two basis sets that can be considered rather extreme in this series of investigations. These basis sets are displayed in Table I. The second one contains a more concentrated set of trigonometric functions giving an extra dense representation in the main resonance energy region between 65 —70 eV. The size of the spin-restricted full configuration Hamiltonian over these basis sets is 2241, (2532 ). The primitive eigensets were obtained from diagonalization for the first 60 roots. A11 calculations were performed at the Tz equilibrium distance of 1.4 a.u. , with no regard to rotational-vibrational excitations or nuclear effects.
RESULTS AND DISCUSSION Resu1ts for discrete-state energy levels of HeT+ and the corresponding population probabilities following P
(8)
decay of Tz are displayed in Table II, together with data from previous work. ' It is emphasized that the presently employed one-center expanded basis set is comparatively more favorable for higher members of the discrete Rydberg series and, especially, in the continuum. Yet the agreement for the lowest states with the results of Kolos et al. , who use explicitly correlated basis functions expanded in elliptical coordinates, and with Martin and Cohen, who use configuration interaction with twocenter Gaussian basis functions, is promising. For the first six states these authors report a cumulative probability of 0.8464 (Ref. 3) [0.8458 (Ref. 8)], while ours is 0.8458. We obtain a total discrete-state probability of 0.8611 to be compared with 0.8573, 0.8575, and 0.8458. The second basis set, with functions more concentrated to the resonance region, predicts a total discrete-state probability of 0.8507, and gives, as expected, individual discrete-state population probabilities more at variance with the accurate results of Kolos et al. The results from Stieltjes imaging for the continuum spectrum from 45 to 90 eV are displayed in Figs. 1(a) and 1 (b}. The position and strengths of the primitive spectrum and the Stieltjes derivative points for different dimensions of the principal representation (pseudospectra) are also given for comparison. As seen in the figure we obtain a "clean" continuum except for one broad and intensive resonance centered at 68 eV, and an excellent fitting to the different pseudospectra. The total continuum cross section is predicted to 0. 1314 (0. 1386 for basis set 2), giving a total cross section of 0.9928 (0.9893 for basis set 2) over the energy range included in the calculations. The results obtained from basis set 2 give a somewhat improved resolution in the Stieltjes imaging and predict structure in the big resonance while the scattering background in the low-energy region is enhanced. The changes in the spectrum between the two calculations indicate that there is a significant interaction between the resonant states and the scattering background and that there is an intensity borrowing among discrete and scattering states. Our values should be compared with those of the two previous investigations on the continuous spectrum of HeT+ following P decay of Tz. Recently Froelich et al. obtained positions and widths for the three lowest reso-
'"
"
CONTINUOUS ELECTRONIC SPECTRUM ACCOMPANYING
38
THE. . .
2709
TABLE I. One-center expanded basis sets (B1 and B2) containing Slater and trigonometric functions. X„~ ((, k, r, 8, $)= A„((,k)r" 'e t"cos(kr)YI (8, $), where n, l, m, denote, respectively, principal, impulse moment, and magnetic quantum numbers; g, k denote the Slater exponents and the trigonometric arguments; and S specifies the c~„irreducible representation. Lines not preceded by a label are common to both basis sets.
1.95 1.580 3.150 6.200
0.000 0.000 0.000 0.000 0.000 0.000
0.5 0.2
0.5+0.2I, 0.9+0. 1I,
1.500 1.580 3. 150 6.200
0.000 0.000 0.000 0.000
0.5 0.2
0.5+0.2I, 0.9+0. 1I,
2.250 1.580 3. 150 6.200
0.000 0.000 0.000 0.000
0.5 0.2
0.5+0.2I, 0.9+0. 1I,
3.000 1.580 3. 150 6.200 1.580 3. 150 6.200 5.040 3.780 5.040 4.410 5.040 1.500 1.580 3. 150 6.200 1.500 1.580 3. 150 6.200
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.50 0.95
B1: 1 B2:
1
B1: 1 B2:
1
B1: 1 B2:
1
0 0 0 0 0 0 0 0 0 0 0 0
—1 —1 —1 —1 1 1
1 1
nance states using a method with analytic continuation of real stabilization graphs. Two strong and narrow resonances were found in the 65 —70-eV energy region, thus overlapping the broader resonance structure we obtain in the present work. These resonances collect much of the total continuum cross section at the expense of the scattering background, which is considerably lower than predicted in this work. We emphasize that in the present approach we obtain primitive energies and probabilities
I =1 to I =1 to
6
I =1 to I =1 to
6
I =1 to I =1 to
6
7
7
7
from diagonalization of the full two-electron Hamiltonian (full configuration interaction CI). This means that the characterization of resonances as closed- ("shape" ) type and open (Feschbach) type is simultaneously accounted for in the accompanying Stieltjes calculation. This means also that the sometimes arbitrary separation of the resonances as open or closed channels is not necessary and that we obtain width and intensity of the resonances and the whole scattering background in one ansatz. The
HANS AGREN AND VINCENZO CARRAVETTA
shape of the resonance and the resolution that can be obtained ultimately depends on the richness of the basis set representation in the resonance region and on the number of pseudospectra (principal representations) that can be used with numerical stability in the Stieltjes-imaging pro-
cedure.
Martin and Cohen were the first to produce a continuum spectrum of HeT+ following the beta decay of T2, and used a Stieltjes-imaging procedure similar to the one used here. They also predicted the large resonance in the 65 —70-eV energy region. The main difference to their work lies in the type of basis set employed. With Gauss-
o np=18 x Ap17 np ~O
0
16 5 4
3 2
C9 C5
1
0 CL
I
20
lI
«
li
40
I
80
60
Energy (eV)
x50
0
O
C9 lO
JD
p
Q
20
40
60
80
Energy (eV)
FICi. 1. Results from Stieltjes imaging of the electromc spectrum of HeT+ following Il decay of T2. The vertical bars represent the probabilities of the original discretized spectrum. The values of the cross sections are marked by different symbols depending on the number (np) of moments considered. The solid curve represents the Stted values for all pseudospectra from np =18 to np =9. Continuum spectrum scaled )& 50. Discrete states overlapping continuum scaled & 10. (a) Basis set 1 (B1). (b) Basis set 2 (B2).
CONTINUOUS ELECTRONIC SPECTRUM ACCOMPANYING
38
THE . .
2711
~
TABLE II. Discrete-state energy levels (eV) and population probabilities of 'HeT+ following P decay of Tz. Probability
Energy
This work
This work
B1
82
Ref. 3
Ref. 8
0.0 27.42 33.82 38.09 39.04 39.70 42.02 42.67 43.82 44.41
0.0 27.49 33.80 39.34 39.80
0.0 27. 35 33.99 38.08 38.98 39.89
0.0 27.29 33.89 37.96 38.82 39.77
41.32
This work
This work
Ref. 11
81
82
0.0 27.02 33.54 37.57 38.42 39.38
0.5832 0. 1645 0.0810 0.0091 0.0008 0.0082 0.0078 0.0007 0.0002 0.0056
0.5857 0. 1604 0.0810 0.0151 0.0055 0.0031
ians it is harder to obtain a wide nonredundant representation of the continuum and at the same time a sufficient richness at a resonance, as can be obtained with basis functions that explicitly contain the oscillating character of the continuum waves. States with spuriously high intensities may thus appear that collect the intensity of a "remainder" Rydberg series not well described by the basis set. We believe that the enhanced background just above the continuum threshold is due to such a basis-set limitation. The shape of the big resonance predicted by Martin and Cohen is rather similar to what we obtain with the enriched basis set in this energy region. In Fig. 2 we display Kurie plots for different values of the neutrino rest mass. It is clear from these plots that
I
-120
FIG. 2. Kurie plots of P decay from
-100
Tz corresponding
0.5823 0. 1681 0.0788 0.0080 0.0001 0.0092
Ref. 8
Ref. 11
0.5822 0. 1675 0.0787 0.0081 0.0001 0.0092
0.5859 0. 1633 0.0770 0.0084 0.0001 0.0090
the effect of different masses is not as significant in the energy range corresponding to the electronic continuum as in the near end-point region, for which the probability distribution of the first few states also should include rotational-vibrational excitations. Kolos and coworkers find that the form of the high-energy tail is not negligible for neutrino masses in the order of 1 eV, but states also that the asymptotic behavior of the probability distribution for the hydrogen atom mimics the true highenergy tail sufficiently well. In order to estimate the sensitivity of the Kurie function on the continuum probability distribution closer to the threshold we show in Fig. 3 differential Kurie plots for two values of the neutrino mass (corresponding numbers are listed in Table III). At
I
-80 -60 E-Ema (eV)
1
-40
-20
to the neutrino masses 0, 15, and 30 eV. Basis set
1
(Bl).
HANS AGREN AND VINCENZO CARRAVEI I A
2712
I
I
I
I
I
-125
-100
-75
-50
-25
E - Emax (eV)
m„=30eV
-160
-140
-100
-120
-80
-60
-40
Emax ~e~~ an thee Kurie plot B2 obtained from basis set 2. (a) Neutrino mass = 5 eV. (b) uric plots. lots A, B,. C,. Dand FIG. 3. Difference between Kurie res o e . A,, B2 spec Neutrino mass = 30 eV. spectrum with all continuum probabilities collecte d at tthreshold. B, B2 spectrum with all continuum probabilities neg l ecte d.. C , B2 d is crete spectrum, 81 continuum spectrum. D, B1 fu spec rum.
100 eV from the end point, the difference of the Kurie function values for the two basis sets is a small fraction of an electron volt, while neglect of the continuum robabilities or addition of all probabilities just at the threshold introduces a deviation of the order o eV. These difference values are small compared to the
— —
absolute function value. However, depending on the performance in terms of signal to noise and resolution in the different energy regions of the experiments yet to be conducted, the continuum portions off the Kurie plots may still be crucial for a determination of a possible neutrino rest mass.
CONTINUOUS ELECTRONIC SPECTRUM ACCOMPANYING
THE. . .
2713
TABLE III. Difference between Kurie plots A, B, C, D and the Kurie plot B2 obtained from basis set 2. 0, full Kurie plot from basis set B2. A, B2 spectrum with all continuum probabilities collected at threshold. B, B2 spectrum with all continuum probabilities neglected. C, B2 discrete spectrum, B1 continuum spectrum. D, B1 full spectrum. Energy
—5.00 —10.00 —15.00 —20.00 —25.00 —30.00 —35.00 —40.00 —45.00 —50.00 —55.00 —60.00 —65.00 —70.00 —75.00 —80.00 —85.00 —90.00 —95.00 —100.00 —105.00 —110.00 —115.00 —120.00 —125.00 —130.00 —135.00 —30.00 —35.00 —40.00 —45.00 —50.00 —55.00 —60.00 —65.00 —70.00 —75.00 —80.00 —85.00 —90.00 —95.00 —100.00 —105.00 —110.00
—115.00 —120.00
—125.00 —130.00 —135.00 —140.00 —145.00 —150.00 —155.00 —160.00
Neutrino
0.0000 7. 1220
11.1465 15.0611 18.9383 22. 7980 26.7744 30.8968 35. 1450 39.4784 43.8744 48.3217 52. 8119 57.3461 61.9313 66.5649 71.2411 75.9543 80.6989 85.4695 90.2619 95.0729 99.8997 104.7403 109.5927 114.4555
119.3275
19.2238 24. 8965 29.7323 34.2254 38.5350 43.4897 48.5008 53.4902 58.3946 63.2469 68.0772 72.9026 77.7695 82.7131 87.6771 92.6459 97.6154 102.5819 107.5426 112.5014 117.4598 122.4184 127.3774 132.3370 137.2971 142.2579
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 —0.6466 —0.9658
—1.2331 —1.4298 —1.5318 —1.6007 —1.6557 —1.7029 —1.7477 —1.7937 —1.8379 —1.8796 —1.9184 —1.9546 —1.9884 —2.0198 —2.0492
5 eV
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0070 0.0239 0.0528 0. 1018 0. 1832 0.2983 0.4438 0.6166 0.8126
—1.0934 —1.2124 —1.3169 —1.4094 —1.4919 —1.5659 —1.6325 —1.6928 —1.7477 —1.7978 —1.8437 Neutrino
0.0000
mass
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 —0. 1296 —0.2795 —0.4535 —0.6348 —0.8048 —0.9578
1.0277 1.2589 1.5037 1.7601 2.0265 2.3016 2. 5842 2.8735 mass
30 eV 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0415
0. 1048 0. 1915 0.3399 0.5803 0.8531 1.1401 1.4358 1.7349 2.0338 2.3354 2.6407 2.9496 3.2621 3.5778 3.8965 4.2179
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0005 0.0014 0.0030 0.0062 0.0117 0.0175 0.0255 0.0358 0.0472 0.0595 0.0726 0.0864 0. 1008 0. 1156 0. 1309 0. 1465 0. 1625
0.0000 0.0149 0.0234 0.0316 0.0397 0.0478 0.0489 0.0490 0.0459 0.0415 0.0303 0.0164 0.0009 —0.0149 —0.0297 —0.0454 —0.0595 —0.0720 —0.0839 —0.0953 —0. 1063 —0. 1168 —0.1269 —0. 1368 —0. 1464 —0. 1557 —0. 1649
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0028 0.0058 0.0107 0.0221 0.0386 0.0451 0.0640 0.0849 0. 1006 0. 1168 0. 1333 0. 1501 0. 1670 0. 1842 0.2015 0.2189 0.2364
0.0000 0.0403 0.0522 0.0623 0.0717 0.0807 0.0559 0.0503 0.0352 0.0163 —0.0178 —0.0438 —0.0664 —0.0818 —0.0917 —0. 1114 —0. 1186 —0. 1237 —0. 1340 —0.1439 —0. 1534 —0. 1626 —0.1716 —0.1805 —0.1892 —0. 1978 —0.2063
2714
HANS AGREN AND VINCENZO CARRAVETTA
ACKNOWLEDGMENTS
We are much obliged to Jan Almlof at the Supercomputer Institute, University of Minnesota, for valuable
'Present address:
Institute of Quantum Chemistry, University of Uppsala, P.O.B. 518, S-75120 Uppsala, Sweden.
Present address: Istituto Chimica Quantistica de Energetica Molecolare del Consiglio Nazionale delle Ricerche, via Risorgimento 35, University of Pisa, 56100 Pisa, Italy. 'W. Kolos, B. Jeziorski, H. J. Monkhorst, and K. Szalewicz, Int. J. Quant. Chem. 19, 421 (1986). 20. Fackler, B. Jeziorski, W. Kolos, H. J. Monkhorst, and K. Szalewicz, Phys. Rev. Lett. 55, 1388 (1985) 3W. Kolos, B. Jeziorski, K. Szalewicz, and H. J. Monkhorst, Phys. Rev. A 31, 551 (1985). 4B. Jeziorski, W. Kolos, K. Szalewicz, and O. Fackler, Phys. Rev. A 32, 2573 (1985).
38
support and for the excellent research facilities put at our disposal. We also thank Hans-Jgrgen Jensen for help with implementation of the determinant-based configuration-interaction program, and Piotr Froehlich for valuable discussions.
5K. Szalewicz, O. Fackler, B. Jeziorski, W. Kolos, and H. J. Monkhorst, Phys. Rev. A 35, 956 (1987). 6P. Froelich, K. Szalewicz, B. Jeziorski, W. Kolos, and H. J. Monkhorst, J. Phys. B (to be published). 7W. Kolos, B. Jeziorski, J. Rychlewski, K. Szalewicz, H. J. Monkhorst, and O. Fackler (unpublished). R. L. Martin and J. S. Cohen, Phys. Lett. 110A, 95 (1985). H. Agren, and V. Carravetta, J. Phys. Chem. 87, 370 (1987). ~OV. Carravetta, and H. Agren, Phys. Rev. A 35, 1022 (1987). "O. Fackler and N. Winter (unpublished) (quoted in Ref. 3); O. Fackler, M. Mugge, H. Sticker, N. Winter, and R. Woerner, in Massive Neutrinos in Astrophysics and in Particle Physics, edited by J. Tran Than Van (La Plagne-Savoie, France, 1984).
